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# Nanophotonics

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# Quantum theory of surface-enhanced resonant Raman scattering (SERRS) of molecules in strongly coupled plasmon–exciton systems

Tomáš Neuman
/ Javier Aizpurua
• Centro de Física de Materiales CFM–MPC, Centro Mixto CSIC-UPV/EHU, 20018 San Sebastián/Donostia, Basque Country, Spain
• Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain
• Other articles by this author:
/ Ruben Esteban
• Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain
• IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Basque Country, Spain
• Other articles by this author:
Published Online: 2019-12-20 | DOI: https://doi.org/10.1515/nanoph-2019-0336

## Abstract

Localised surface plasmons can couple strongly with the electronic transitions of a molecule, inducing new hybridised states of light and matter, the plasmon–exciton polaritons. Furthermore, molecules support vibrational degrees of freedom that interact with the electronic levels, giving rise to inelastic resonant Raman scattering under coherent laser illumination. Here we show the influence of strong plasmon–exciton coupling on resonant Raman processes that populate the vibrational states of the molecule and that lead to the characteristic surface-enhanced Raman scattering spectra. We develop analytical expressions that give insight into these processes for the case of moderate illumination intensity, weak electron–vibration coupling and no dephasing. These expressions help us to elucidate the twofold role of plasmon–exciton polaritons to pump the system efficiently and to enhance the Raman emission. Our results show a close analogy with the optomechanical process described for off-resonant Raman scattering but with a difference in the resonant reservoir. We also use full numerical calculations to study the effects reaching beyond these approximations and discuss the interplay between the fluorescence background and the Raman lines. Our results allow for better understanding and exploitation of the strong coupling regime in vibrational pumping and in the surface-enhanced resonant Raman scattering signal.

## 1 Introduction

Metallic nanoparticles support plasmonic modes at optical frequencies than can confine the electromagnetic energy to effective volumes [1], [2], [3], [4], [5], [6] even below ≈10–1000 nm3 and thus give rise to efficient coupling of the particle plasmons with excitations in nearby molecules (e.g. molecular excitons). This efficient interaction has found, among others, a prominent application in surface-enhanced Raman scattering (SERS) [7], [8], [9], [10], [11], [12], where the inelastic Raman signal containing the vibrational signature of the molecule is enhanced.

The dynamics of the SERS process has been recently described in a cavity Quantum Electrodynamics (QED) framework [5], [13], [14], [15], [16], [17], [18]. In particular, it has been shown that in off-resonance SERS, where the electronic transitions of the molecules are strongly detuned with respect to the exciting laser, the underlying physics can be understood as a molecular optomechanical process [5], [14], [16], [17], [18], [19], [20], [21], in which the vibration of the molecules and the plasmonic resonance replace the macroscopic mechanical vibrations and the optical mode in conventional optomechanics [22], respectively. This QED-based theory has described regimes of nonlinear SERS with the use of a single incident laser and has naturally addressed the effect of optomechanical vibrational pumping [23], [24], [25], [26], [27], a process where enhancement of the Stokes–Raman emission leads to population of the vibrational mode of the molecule.

On the other hand, in surface-enhanced resonant Raman scattering (SERRS), an electronic transition close to the energy of the exciting laser considerably affects the Raman process and gives rise to complex nonlinear phenomena [28]. In this case, it is possible to describe the light–matter interaction by a similar Hamiltonian as the one used in hybrid optomechanics [13], [29], [30], where a two-level excitation is included in the system, leading to a nonlinear behaviour of the response.

The system is said to be in weak-coupling regime (or in the bad-cavity limit), when the interaction strength, g, between the plasmon and the electronic transition remains small in comparison with the plasmonic losses, γa (assuming negligible molecular losses and pure dephasing), i.e. gγa/2. Under such conditions, the electronic transition in the molecule and the plasmon can be considered as well-defined excitations that interact weakly. We have addressed this situation in our previous work [28], where we also compare the model describing resonant Raman with the off-resonant Raman model.

On the other hand, if the coupling between the plasmon and the molecular electronic states is sufficiently large to overcome the plasmonic losses, i.e. gγa/2, the system reaches the regime of strong coupling and the electronic transition hybridises with the plasmon. Large enough values of g to achieve strong coupling are nowadays typically achieved by coupling plasmonic nanostructures with either J-aggregates [31], [32], [33], [34], [35], [36], [37], [38], [39], large complex dyes characterised by large dipole moments, or with a large number of small molecules forming a collective bright mode of larger effective coupling strength [40], [41], [42], [43], [44]. However, recently, it has become possible to achieve strong coupling in plasmonic systems even with a single small molecule or quantum dot [37], [42], [43], [45], [46], [47], opening new possibilities for fundamental studies of hybrid light–matter quantum states. The regime of strong coupling between a single two-level electronic transition and a plasmonic mode is characterised by a structure of energy levels forming the so-called Jaynes–Cummings (J-C) ladder, which leads to complicated patterns of inelastic resonance–fluorescence light emission [48], [49].

In this work, we present a theoretical study of vibrational pumping and Raman scattering in resonant SERS systems [50] in the framework of cavity QED, focusing on the regime of strong coupling between the single molecule and a plasmonic mode. After a brief introduction to the regime of strong coupling, we describe the resonant SERS Hamiltonian and develop a simplified, but intuitive, analytical approach that allows for understanding the physical mechanism of vibrational pumping and vibrational Raman scattering. Specifically, we assume a scenario where the illuminating laser is not too strong, the pure dephasing processes that broaden the electronic transitions are not considered, and the coupling between electronic and vibrational levels of the molecule is sufficiently weak so that the zero-phonon line dominates the fluorescence spectrum. In the second part of the article, we numerically solve the system’s dynamics without simplifications to show successively how large electronic–vibrational coupling, pure dephasing, and intense laser illumination, effects that might be important in real experiments, affect the response of the system.

## 2 Plasmon–exciton strong coupling

Before discussing the SERS process, we briefly summarise the properties of a plasmonic system strongly coupled with an electronic transition. The system under consideration is sketched in Figure 1A, where a plasmonic structure resonant at energy ωpl interacts with a molecule, described as a two-level system (TLS) of excitation energy ωeg.

Figure 1:

Strong coupling between a plasmonic system and an electronic transition of a molecule, in the absence of molecular vibrations.

(A) Schematic depiction of the possible realisation of the Jaynes–Cummings (J-C) model. A dye molecule with an excitonic transition of energy ħωeg is placed near a metallic cavity that sustains a plasmonic mode at energy ħωpl. Both systems are tuned to the resonant condition ωeg=ωpl. The plasmonic cavity is illuminated by a single pump laser of frequency, ωL, or by a combination of a pump and a probe laser. (B) J-C ladder showing the coherent mixing of the electronic levels of the molecule with the bosonic levels of the plasmon (defined by states |N, g(e)⟩ in the noninteracting case) into a set of hybrid states |N, ±⟩ under the strong-coupling conditions. The blue frame marks the single-excitation manifold determining the system response under weak-illumination conditions. (C) Absorption spectrum of a weakly perturbing probe for pumping amplitude ħℰ=1 meV. The pump laser frequency is fixed to ωL=ωpl (Δ=0), and we change the frequency of the probe laser. (D) Emission spectrum of the J-C system for zero detuning of the laser ωL=ωpl (Δ=0) and pumping amplitude ħℰ=1 meV. The plasmon–molecule interaction considered for the calculations corresponds to the strong-coupling regime and is described within the J-C model for the following values of the parameters: ħγσ=2×10−5 eV, ħγa=150 meV, ħg=100 meV, ħωpl=ħωeg=ħωL=2 eV.

This system can be described within the rotating wave approximation by the J-C Hamiltonian, and we thus refer in the following to the J-C system when discussing the coupling of the TLS and the plasmon in the absence of any vibration. The J-C Hamiltonian driven by a laser of frequency ωL can be written as follows:

$HJ−C=ℏωpla†a+ℏωegσ†σ+ℏga†σ+ℏg∗aσ†+ ℏℰaeiωLt+ℏℰ∗a†e−iωLt,$(1)

with a(a), the annihilation (creation) bosonic operators associated to the plasmon; and σ(σ), the TLS lowering (raising) operators. Here g is the plasmon–molecule coupling strength, and the pumping by the laser is introduced via the parameter ℰ, proportional to the amplitude of the electric field of the driving laser and to the transition dipole moment of the molecule. We further assume that both g and ℰ are real. The explicit time dependence appearing in the pumping term is transformed out of the Hamiltonian by working in the standard interaction picture where the new Hamiltonian is obtained from the unitary transformation H′=UHU+iħU̇U, with U=exp(iHNIt), and HNI=ħωL(aa+σσ) is the noninteracting part of the Hamiltonian. In this picture, the explicit time dependence of the Hamiltonian drops out, the plasmon frequency is replaced by ωpl→Δ=ωplωL, and the molecule TLS frequency transforms analogously as ωegδ=ωegωL. For simplicity, throughout the article, we consider that the frequency of the molecular transition is exactly tuned to the plasmon frequency ωpl=ωeg (Δ=δ).

Losses are added by using the master equations for the time evolution of the density matrix, ρ, and introducing Lindblad terms for the TLS, ℒσ(ρ), and the plasmon, ℒa(ρ),

$ℒσ(ρ)=−γσ2({σ†σ, ρ}−2σρσ†),$(2)

$ℒa(ρ)=−γa2({a†a, ρ}−2aρa†)$(3)

with γσ and γa, the corresponding loss rates of the exciton and plasmon, respectively, and {·,·}, the anticommutator. We neglect the effect of finite temperature (we consider T=0 K). The possibility of pure dephasing, leading to decoherence of the TLS without directly involving decay from the excited to the ground state, is considered in a later section.

To understand the behaviour of the system, we first briefly consider that the plasmon excitation is switched off. In this case, the system can be represented in the basis of states |N, g(e)⟩=|N⟩⊗|g(e)⟩ (with ⊗ the direct product), combining the plasmon number states |N⟩ and the electronic ground (excited) state |g(e)⟩ of the molecule. For ωeg=ωpl and negligible interaction g, the states |N–1, e⟩ and |N, g⟩ are degenerate, and their energies form an equally spaced ladder (at the left of the scheme in Figure 1B). When g overcomes the system losses, the system enters the strong coupling regime where the degeneracy is lifted, resulting in a new set of hybrid polaritonic states |N, +⟩ and $|N,\text{\hspace{0.17em}}–〉\text{\hspace{0.17em}}\left[|N,\text{\hspace{0.17em}}±〉=\left(|N-1,\text{\hspace{0.17em}e}〉±|N,\text{\hspace{0.17em}g}〉\right)\text{}/\text{}\sqrt{2}\right].$ These states form a set of structured levels called the J-C ladder, which is sketched in Figure 1B to the right of the uncoupled |N, g(e)⟩ levels. The energy splitting between these two states, ΔEJ−C(N), is proportional to the square root of the number of plasmonic excitations present in the system and to the coupling constant g via $\Delta {E}_{\text{J}-\text{C}}\left(N\right)=2\sqrt{N}\hslash g.$ For this general discussion, we ignore that the losses can modify the exact splitting and coherent mixing of the bare states. For weak illumination, N≪1, most transitions occur between the ground state and the so-called single-excitation manifold containing the states |1, ±⟩ (blue frame in Figure 1B), so that the absorption peaks appear at ≈ωpl±g.

To illustrate this behaviour, we first consider that the system is pumped by a laser at frequency ωL in (1) and probed by a perturbative laser with frequency ω [note that the probe is not explicitly present in the Hamiltonian of (1)]. The system is strongly coupled with the parameters ħg=100 meV and ħγa=150 meV. We calculate the absorption1 as the Fourier transform of the two-time correlation function:

$sA(ω)∝−2Re∫0∞〈[a†(0), a(τ)]〉ss eiωτd τ≈2Re∫0∞〈a(τ)a†(0)〉ss eiωτd τ,$(4)

where [·,·] denotes the commutator, and ⟨·⟩SS is the steady-state mean value, which can be evaluated with help of the quantum regression theorem [51].

In Figure 1C, we plot sA(ω) for a pumping amplitude ħℰ=1 meV that weakly populates the system. The results in Figure 1C are in good agreement with the expectation foreseen from the structure of the J-C ladder. The absorption spectrum consists of two absorption peaks separated by ≈200 meV≈2ħg emerging from the transitions |0, g⟩→→|1, ±⟩, in accordance with our discussion of the hybrid polaritonic states.

To facilitate the understanding of the following sections, we also briefly introduce the emission of the pumped J-C model. In this case, we excite the system with a monochromatic laser and study the emitted spectra given by the one-sided Fourier transform of the two-time correlation function:

$sE(ω)∝2Re∫0∞[〈a†(0)a(τ)〉ss − 〈a†(0)〉SS〈a(τ)〉SS] eiωτd τ.$(5)

We plot in Figure 1D the inelastic emission spectra (for clarity, we always subtract the elastic component) of the J-C system for a pumping amplitude ħℰ=1 meV and for ħωL=ħωpl=ħωeg=2 eV. The emission spectrum yields a double-peak structure originated from the incoherent decay of the polaritonic states to the ground state (|1, ±⟩→|0, g⟩).

Last, we emphasise that here we focus on the weak illumination regime, in which the laser moderately populates the polaritonic states (N≪1). For intense laser illumination, the optical absorption and light emission from a strongly coupled J-C system are nontrivial, as described in detail elsewhere [48], [52]. We briefly discuss this regime of strong illumination in a separate section at the end of this work.

## 3 Quantum description of SERRS in strong coupling

We model the plasmon-enhanced vibrational Raman processes as sketched in Figure 2A. We consider a single vibrational mode of the molecule that, in the harmonic approximation, is described as a bosonic excitation of angular frequency ωm, with b (b) its annihilation (creation) operators. The Hamiltonian of the uncoupled vibrations is

Figure 2:

Schematics of the system considered for the description of SERRS alongside with examples of inelastic SERRS and Jaynes-Cummings spectra.

(A) Schematic depiction of the system where a plasmonic mode interacts with the electronic states of the molecule, which are further coupled to molecular vibrations. A dye molecule with an excitonic transition of energy ħωeg is placed into a plasmonic cavity that supports a resonance at energy ħωpl tuned to the molecular transition (ħωpl=ħωeg). Each molecular electronic level serves as a potential energy surface (PES) supporting a vibrational mode of energy ħωm. The excited state and the ground state PES are mutually displaced as given by the dimensionless displacement d. The plasmon is illuminated by a single intense laser of frequency ωL and pumping amplitude ℰ that triggers the surface-enhanced Raman scattering process. (B) Two examples of emission spectra calculated using the full model where a vibrational mode is present (black lines) and the J-C model without considering the vibrations (red dashed line) calculated for ħℰ=1 meV and 10 meV, respectively. The spectra in (B) are calculated using the parameters for the J-C model as in Figure 1 and the following values for the vibrations: ħγb=2 meV, ħωm=50 meV, and $\sqrt{S}=0.1.$

$Hvib=ℏωmb†b.$(6)

This vibrational mode interacts with the electronic TLS of the molecule via a polaronic coupling characterised by a dimensionless displacement $d=\sqrt{S}$ (S is also known as the Huang–Rhys factor) [50], [53], [54], [55]:

$Hpol=ℏωmdσ†σ(b†+b).$(7)

The total Hamiltonian thus becomes

$H=HJ−C+Hvib+Hpol,$(8)

with HJ−C given by (1). As before, we consider that the plasmon resonance is tuned to the electronic transition of the molecule, ωpl=ωeg. Furthermore, besides the losses of the plasmon and the TLS described in the previous section, we incorporate decay processes of the vibrational states via the Lindblad term ${ℒ}_{b}\left(\rho \right)=-\frac{{\gamma }_{b}}{2}\left(\left\{{b}^{†}b,\text{\hspace{0.17em}}\rho \right\}-2b\rho {b}^{†}\right),$ with γb the decay rate of the vibrations.

To illustrate the resonant Raman spectra, we show in Figure 2B the inelastic emission spectra [following (5)] calculated within this full model (black lines) and also for the J-C model neglecting vibrations [(1), red dashed line]. The spectra are plotted for two different strengths of the laser, corresponding to moderate pumping (ħℰ=1 meV) and to excitation near the continuous-laser damage threshold of the sample (ħℰ=10 meV) [5]. In the absence of vibrations, we obtain in both cases the double peak structure that is characteristic of the J-C Hamiltonian before high population states are significantly excited (Figure 1C, D). The main effect of including the vibrations is the emergence of the Stokes line at energy ħωLħωm, on top of the J-C (background) spectrum. We also find an anti-Stokes peak at ħωL+ħωm, which is much weaker than the Stokes line, as expected for typical vibration populations ⟨bb⟩≪1. A second-order Stokes peak can be also observed at ħωL–2ħωm for ħℰ=1 meV. Further, for ħℰ=10 meV, as compared to the ħℰ=1 meV case, the background shows qualitatively similar features, but it is four orders of magnitude stronger, i.e. grows as ∝|ℰ|4. On the other hand, the maximum of the SERS Stokes line grows approximately linearly with the laser intensity ∝|ℰ|2 as long as the incoherent vibrational population is small. This different scaling with laser intensity decreases the spectral contrast of the Raman signal for the stronger laser, although we observe that for the parameters considered here the main Stokes line clearly dominate the spectrum. We address the strong-intensity effects that go beyond the simple intensity scaling law at the end of the paper.

## 4 Analytical model

To provide insight into the scattering of light by a molecule, we develop an analytical model describing the Raman process for a relatively simple scenario. We first consider that the electron–phonon coupling, described by the displacement value $d=\sqrt{S},$ is in the low range of possible values, as obtained for rigid molecules [56], [57] (d~10−1). We further assume that the laser illumination is not too intense so that the linear response approximation is justified and that the Two-level system is not affected by pure dephasing processes.

The condition of weak electron–vibration coupling, considering that the time scale of the electron–vibration interaction is larger than that of the typically dominant relaxation processes, can be roughly estimated as ωmd<γa/2, g. This assumption allows us to separate the full system according to the hierarchy of time scales: on the one hand, the fast-decaying and decohering reservoir part, which consists of the TLS strongly coupled with the plasmon, and on the other hand, the slowly varying part of the system represented by the vibrational mode [51]. This separation of time scales allows for using the Markov approximation and obtaining the effective vibrational dynamics under the influence of the reservoir. The derivation is discussed further in the Supplementary Information and yields the following set of equations:

$ddt〈b〉=−iωm〈b〉−iωmd〈σ†σ〉SS −(γb/2+Γ−/2−Γ+/2)〈b〉,$(9)

$ddt〈b†b〉=−iωmd〈σ†σ〉SS(〈b†〉−〈b〉)− (γb+Γ−−Γ+)〈b†b〉+Γ+,$(10)

where Γ+ and Γ are, respectively, the effective incoherent pumping and damping rates due to the reservoir. These rates are related to the steady-state reservoir spectral function $\mathcal{J}\left(s\right)=\mathrm{Re}\left\{{\int }_{0}^{\infty }〈〈{\sigma }^{†}\sigma \left(\tau \right){\sigma }^{†}\sigma \left(0\right)〉〉{e}^{\text{i}s\tau }\text{d}\tau \right\}$ as follows:

$Γ−=2ωm2d2J(ωm)$(11)

$Γ+=2ωm2d2J(−ωm),$(12)

with the following notation ⟨⟨O1O2⟩⟩=⟨O1O2⟩–⟨O1SSO2SS. In the scheme considered above, the reservoir operators are obtained from the dynamics of the J-C system decoupled from the vibrations.

$〈b〉SS=−iωmd〈σ†σ〉SSγb/2+Γ−/2−Γ+/2+iωm,$(13)

$〈b†b〉SS=|〈b〉SS|2+Γ+γb+Γ−−Γ+,$(14)

As we have assumed that the pure dephasing of the TLS is negligible, we can write ⟨σσSS≈|⟨σSS|2. Next, we decompose the lowering operator of the TLS into its steady-state value ⟨σSS and the fluctuating part with zero mean δσ, as σ=⟨σSS+δσ, and thus we can approximate the population operator as σσ≈|⟨σSS|2+⟨σSSδσ+σSSδσ.

We then transform the dependency of the spectral function J(s) from the correlation function of the full operator ⟨⟨σσ(τ)σσ(0)⟩⟩ to a simpler dependence with the correlation function ⟨⟨δσ(τ)δσ(0)⟩⟩=⟨⟨σ(τ)σ(0)⟩⟩:

$J(s)=Re {∫0∞〈〈σ†σ(τ)σ†σ(0)〉〉eisτdτ}≈ |〈σ〉SS|2Re {∫0∞〈〈σ(τ)σ†(0)〉〉ei(s+ωL)τdτ}= |〈σ〉SS|2g2γa2+γσ2[γa24 + (Δ − s)2]g4−2g2[(Δ−s)2−γσγa4]+[γσ24+(Δ−s)2][γa24+(Δ−s)2]︸≡J0(s).$(15)

Equation (15) thus can be factorised into two contributions: (i) the coherent population of the TLS, |⟨σSS|2, which in the limit of low illumination intensity is given by the following expression:

$|〈σ〉SS|2≈〈σ†σ〉SS≈g2|α|2(Δ2+γa2/4)Δ4+(g2+γσγa/4)2+Δ2(γσ2/4+γa2/4−2g2),$(16)

with $\alpha =\frac{-ℰ}{\Delta -\text{i}{\gamma }_{a}/2},$ and (ii) the absorption spectrum of the TLS (that contains the coupling of the TLS with the plasmon), J0(s). An approximate version of the latter can be obtained analytically using the quantum regression theorem and considering the low pumping regime. The details of this derivation are given in the Supplementary Information, and the final result appears as the last expression in (15).

In the strong coupling regime, J0(s) contains two peaks (i.e. maxima) approximately located at frequencies ${s}_{1}^{\mathcal{J}}\approx \Delta +\sqrt{{g}^{2}-\left({\gamma }_{\sigma }^{2}+{\gamma }_{a}^{2}\right)/8}$ and ${s}_{2}^{\mathcal{J}}\approx \Delta -\sqrt{{g}^{2}-\left({\gamma }_{\sigma }^{2}+{\gamma }_{a}^{2}\right)/8}$ (assuming a slowly varying numerator). We notice that the double-peaked structure can be resolved only if ${g}^{2}>\left({\gamma }_{\sigma }^{2}+{\gamma }_{a}^{2}\right)/8.$

The expressions for the effective damping (11) and pumping (12), proportional to J(s), can then be interpreted as being the consequence of a two-step process. First, |⟨σSS|2 indicates that the pumping laser of frequency ωL induces coherent population of the molecular excited state, which triggers the Raman process and thus increases Γ and Γ+. In a second stage, the Stokes and anti-Stokes excitations appearing at frequencies ωL±ωm are absorbed and partially emitted to the far field via the resonance of the J-C system, given by J0(s) [the emission at the Raman frequencies corresponds to J0ωm)]. The enhancement of the Stokes scattering at frequency ωLωm leads to the enhancement of the vibrational pumping Γ+ (a vibrational quantum is created with each emitted Stokes photon), while the anti-Stokes emission at frequency ωL+ωm contributes to the vibrational damping Γ. A similar interpretation has also been shown to be valid for off-resonant SERS [17], [18], [58], [59].

Figure 3 demonstrates the validity of this description. We show in Figure 3A and B two-dimensional D colour-map plots of the incoherent part of the vibrational population, ⟨⟨bb⟩⟩SS=⟨bbSS–|⟨bSS|2, as a function of the vibrational frequency ωm and of the detuning Δ of the laser from the plasmon frequency. We display the incoherent part of the population as the coherent part |⟨bSS|2 represents a constant static displacement of the vibration due to the coherent laser pumping.

Figure 3:

Resonant surface-enhanced Raman scattering for weak illumination (ħℰ=1 meV), no pure dephasing, and weak coupling between the electronic and vibrational levels of the molecule, d=0.1.

(A, B) Incoherent population of the vibrational mode under laser excitation as a function of the laser detuning Δ and vibrational frequency ωm, calculated (A) from the analytical model and (B) from the full numerical calculation. The lines mark the conditions when the frequency of one of the bare J-C absorption peaks coincides with the frequency of the first-order Stokes line (red line) or the incident laser (green line). (C) Numerically calculated surface-enhanced resonant Raman scattering spectra sE(ω, Δ) at emission frequency ω as a function of the plasmon detuning Δ, for a vibrational mode energy ħωm=50 meV. The black dashed lines mark positions of the first- and second-order Stokes lines. The white dashed lines mark the condition for which either the frequency of the first-order Stokes line or the frequency of the incident laser matches with one of the Jaynes–Cummings peaks. (D) Maximum intensity of the Stokes Raman line (at ω=ωLωm) as a function of the detuning of the incident laser from the plasmon Δ, calculated with the analytical model [(17) (red line)] and with the full numerical model (black crosses). The other parameters used for the calculations in (A–D) are ħγb=2 meV, ħγσ=2×10−5 eV, ħγa=150 meV, ħg=100 meV, ħγϕ=0 eV, Δ=δ.

The approximate (Figure 3A) and exact (Figure 3B) results are very similar, both exhibiting a clear population maximum for a detuning ħΔ≈–100 meV and a frequency of the vibrational mode ħωm≈180 meV. This maximum can be understood as a consequence of the two-step process described above. Pumping, |⟨σSS|2, is optimised when the incident laser excites the absorption peaks near the resonant condition ωLωpl±g (ωLωpl±g marked by the vertical green line). On the other hand, the Raman–Stokes process, leading to vibrational pumping, is maximised when the Stokes–Raman line coincides with one of the J-C absorption peaks (red lines, marking the condition ωLωmωplg). Figure 3A and B show that the strongest incoherent vibrational population is indeed achieved when both conditions are fulfilled (region near the intersection of the green and the red line). We remark that for the regime under consideration, ħℰ=1 meV and temperature T≈0 K, the enhancement of the anti-Stokes lines does not significantly influence the incoherent vibrational population.

Although the incoherent vibrational populations provide important information about the system dynamics, the quantity typically accessible in experiments is the inelastic Raman emission spectrum [given by (5)]. Using the separation of the system’s dynamics into the slow vibration and the fast J-C reservoir, we can isolate the Raman contribution to the scattering spectrum from the fluorescence background (as discussed in Supplementary Information). By doing this, the resulting expressions for the Stokes (se,S) and anti-Stokes (se,aS) emission lines are as follows:

$se,S(ω)=ωm2d2|〈σ〉SS|2|A(−ωm)|2 × 2Re{∫0∞〈〈b(0)b†(τ)〉〉ei(ω−ωL)τdτ},$(17)

$se,aS(ω)=ωm2d2|〈σ〉SS|2|A(ωm)|2 × 2Re{∫0∞〈〈b†(0)b(τ)〉〉ei(ω−ωL)τdτ},$(18)

where |⟨σSS|2 is given by (16), and |A|2 is the reservoir spectral function that describes the enhancement of the emission of the Raman photons into the far field due to the presence of the J-C system (we assume that the emission from the molecule via the plasmon is much stronger than its direct far-field emission):

$|A(s)|2=g2g4−2g2[(Δ−s)2−γσγa/4]+[γσ2/4+(Δ−s)2][γa2/4+(Δ−s)2].$(19)

This function is double peaked, and for the conditions considered in this article, the peak positions, ${s}_{1}^{\mathcal{A}}=\Delta +\sqrt{{g}^{2}-\left({\gamma }_{\sigma }^{2}+{\gamma }_{a}^{2}\right)/8}$ and ${s}_{2}^{\mathcal{A}}=\Delta -\sqrt{{g}^{2}-\left({\gamma }_{\sigma }^{2}+{\gamma }_{a}^{2}\right)/8},$ are similar as ${s}_{1}^{\mathcal{J}}$ and ${s}_{2}^{\mathcal{J}}$ for the spectral function J0(s).

Last, the integrals of the vibrational correlation functions in (17) and (18) represent the emission line shape that can be obtained from the equations describing the dynamics of the vibration coupled with the reservoir [17] (9, 10). Applying the quantum regression theorem to these equations, we obtain

$Re{∫0∞〈〈b(0)b†(t)〉〉ei(ω−ωL)t} =γvdp/2(ω−ωL+ωm)2+γvdp2/4(1+〈〈b†b〉〉SS),$(20)

$Re{∫0∞〈〈b†(0)b(t)〉〉ei(ω−ωL)t} =γvdp/2(ω−ωL−ωm)2+γvdp2/4〈〈b†b〉〉SS,$(21)

with γvdp=γb–Γ+. The emitted signal (17 and 18) is thus proportional to the efficiency of the excitation of the system, ∝|⟨σSS|2, and to the emission enhancement due to the coupled TLS plasmon, |A(s)|2. Furthermore, the anti-Stokes and Stokes lines are proportional to the incoherent population of the vibrations ⟨⟨bb⟩⟩SS and to 1+⟨⟨bb⟩⟩SS [60], respectively, where ⟨⟨bb⟩⟩SS itself is enhanced by the effect of the reservoir on the excitation and emission process (11–15). The line width of the Raman peaks is finally given by the interplay between the damping rates, γb together with Γ, both broadening the peak, and the pumping rate Γ+ that narrows the peak. These properties again mirror the behaviour that has been discussed for off-resonant Raman [16], [17], [18], except for the existence of a more complex reservoir.

Figure 3C and D illustrate the behaviour of the resulting Raman emission for a vibrational mode of energy ħωm=50 meV and $d=\sqrt{S}=0.1.$ We first show in Figure 3C the emission spectra obtained from the numerical solution of the full system [Hamiltonian in (8) and corresponding loss terms], as a function of the detuning Δ of the laser from the plasmon frequency. The inelastic emission can be split into two components, the emission resulting from the resonance fluorescence of the J-C system and the Raman emission yielding the vibrational lines. The former is present in the emission spectra of Figure 3C as a broad background symmetrical around the zero detuning ħΔ=0 meV, in agreement with previous studies [28], [48]. The strong Raman–Stokes line is also clearly distinguished on top of the background at ωLωm, accompanied by a second-order Stokes transition line (not described by the analytical model that accounts only for the first-order transition) at ωL–2ωm and a very weak anti-Stokes line at ωL+ωm.

To better analyse the Δ dependence of the Raman signal and further test the validity of our model, we focus in Figure 3D on the maximum of the Stokes line, which we plot as a function of the laser detuning Δ as calculated from the full numerical model (black dots) and as given by the analytical expression in (17) (red line). The agreement between the two is excellent. We observe a double-peaked structure that is, however, not symmetric with respect to Δ=0, but with the central minimum blue-detuned by 25 meV. This blue detuning can be understood as the result of having to optimise the product of the enhancement at the excitation ωL and emission ωLωm frequencies, as given by |⟨σSS|2 and |A(s)|2, respectively (assuming that the incoherent vibrational population remains small ⟨⟨bb⟩⟩SS≪1). The individual contributions of |⟨σSS|2 and |A(s)|2 are shown in Figure S1 of the Supplementary Information.

## 5 Effect of strong electron–vibration coupling

We now consider scenarios that go beyond the assumptions of the previous section. Up to now, we have assumed that the coupling between the molecular vibrational and electronic states, characterised by the dimensionless displacement d (the Huang–Rhys parameter S=d2), was weak. However, when the interaction becomes stronger, and thus the time scale of the coupling between the electronic TLS and the phonon is shortened, the dynamics of the J-C system coupled to the vibrational mode are no longer separable as a slow vibrational dynamics driven by the fast relaxing reservoir. For large d, we thus cannot apply the description developed in the previous section, and we use full numerical solutions to obtain the response of the system.

We show in Figure 4A and B the maps of the incoherent vibrational population for large electron–vibration couplings (A) d=0.5 and (B) d=1 (B) as a function of laser detuning and vibrational frequency. The lines in Figure 4A and B have the same meaning as in Figure 3A and B, indicating the detuning Δ at which the first-order Stokes line (red line) or the incident laser frequency (green line) is tuned to the position of one of the absorption peaks of the J-C system. For d=0.5 (Figure 4A), the strongest vibrational pumping appears approximately when both conditions are met and the lines cross, similarly as for low d=0.1 (Figure 3A and B). Nonetheless, while for d=0.1 a single clear peak was observed, two close maxima start to emerge near this optimal condition for d=0.5. The effect of large electron–vibration coupling is more apparent in the case with d=1 (Figure 4B). In this situation, the dependence of the incoherent vibrational population on the vibrational frequency and detuning becomes very complex as it involves the influence of higher-order vibronic transitions.

Figure 4:

Effect of electron–vibration coupling on resonant surface-enhanced Raman scattering (SERS).

(A, B) Incoherent population of the vibrational mode as a function of the laser detuning Δ and vibrational frequency ωm for an electron–vibration coupling (Huang–Rhys) parameter (A) $d=\sqrt{S}=0.5$ and (B) d=1. Results are obtained numerically. In both cases, ħℰ=1 meV, and there is no pure dephasing. The straight lines mark the conditions when the peak frequency of one of the bare J-C absorption peaks coincides with the frequency of the incident laser (green line) or with the first-order Stokes line (red line). (C, D) SERS spectra of a molecule supporting a single vibrational mode of energy ħωm=50 meV, for two values of the electron–vibration coupling, (C) d=0.5 and (D) d=1, and illumination intensity ħℰ=1 meV and 10 meV. The other parameters used for all the calculation are as follows: ħγb=2 meV, ħγσ=2×10−5 eV, ħγa=150 meV, ħg=100 meV, ħγϕ=0 eV, Δ=δ.

Next we consider the emission spectra for Figure 4C moderate (d=0.5) and Figure 4D strong (d=1) electron–vibration coupling when two values of pumping amplitude, ħℰ=1 meV and 10 meV, are considered. The spectra for weak electron–vibration coupling (d=0.1) are shown in Figure 2B. For d=0.5, the results for both illumination intensities show a clear lower-order Stokes peak at ω=ωLωm and a weaker anti-Stokes peak at ω=ωL+ωm. A series of higher-order Stokes peaks is observed at ω=ωLm (with n=2, 3, 4), much stronger than in the case of the spectra calculated for d=0.1. Increasing the displacement d of the excited state vibrations naturally leads to larger overlaps among vibrational wave functions belonging to different numbers of vibrational excitations, thus facilitating higher-order Raman transitions that yield intense higher-order Raman peaks. This can be observed most clearly in Figure 4D, where, for d=1, particularly intense Raman peaks of higher orders emerge on both the Stokes and anti-Stokes sides of the spectra, most notably for ħℰ≈10 meV.

## 6 Effect of pure dephasing

So far, we have assumed negligible dephasing. This is the case for optimised experiments at low temperature, but for a typical situation at finite temperature, the electronic transition is subjected to a large pure dephasing rate, γϕ, due to interactions with the environment. In molecules decoupled from the plasmonic cavity, pure dephasing introduces a loss of coherence between the states of the TLS but without directly inducing the decay of the excited state [49]. A careful analysis of the microscopic dephasing mechanism shows that when the polaritonic states |N, ±⟩ are formed, the interaction of the system with the originally purely dephasing reservoir leads to novel incoherent mechanisms including the energy transfer from the upper polariton states |N, +⟩ to the lower ones |N, –⟩ within the same manifold [61], [62]. In the following, we again consider that the system is illuminated by a weak laser, and hence only the single-excitation manifold containing the states |1, ±⟩≡|±⟩ is important for the description of the inelastic light emission from the polaritons.

We implement the dephasing processes by incorporating the following Lindblad terms in the master equation:

$ℒσ−−++(ρ)=−γϕ({σ−−++σ−−++, ρ}−2σ−−++ρσ−−++)$(22)

$ℒσ−+(ρ)=−γσ−+2({σ−+†σ−+, ρ}−2σ−+ρσ−+†),$(23)

where σ− −++=|+⟩⟨+|+|–⟩⟨–| and σ−+=|–⟩⟨+|. We neglect any further incoherent processes such as the energy transfer from the lower to the upper polariton as we assume that they are less effective [15], [62]. We further set ħγ−+=40 meV whenever we consider ħγϕ≠0 eV, a rate that leads to relatively strong transfer of populations towards the lower polaritonic state via the vibrational reservoir of the molecule [63], [64].

To illustrate the effect of dephasing, we start by plotting the absorption, sA(ω) (Figure 5A), and emission, sE(ω) (Figure 5B), spectra for the bare plasmon–exciton J-C system, without vibrations. In both cases, we set ħℰ=1meV, and we show results for ħγϕ=0 meV, 10 meV, and 50 meV. For clarity, the spectra are vertically shifted. When the dephasing is increased, we observe a clear broadening of the polariton peaks in the absorption spectrum (Figure 5A), which is larger for the upper polaritonic peak due to the population transfer given by terms in (23).

Figure 5:

Effect of pure dephasing for weak illumination (ħℰ=1 meV) and electron–vibration coupling d=0.1.

(A) Absorption spectra and (B) emission spectra for the simple J-C system in Figure 1, displayed for three different values of pure dephasing, ħγϕ=0 meV, 10 meV, and 50 meV, pumped by monochromatic illumination tuned to the plasmonic resonance ħΔ=ħωplħωL=0 eV. (C) Emission spectra of the full molecule–plasmon system (sketched in Figure 2A) including the vibrational mode of frequency ħωm=50 meV, for ħΔ=0 eV and the same γϕ values as in (A, B). The spectra in (A–C) are offset from each other by a constant value of sA=20 in (A) and sE=0.5×10−3 in (B, C). (D) Incoherent populations of the vibrational mode as a function of detuning and vibrational frequency for pure dephasing ħγϕ=10 meV. The lines drawn into the colour maps mark the detuning Δ for which the first-order Stokes line (red line) or the incident laser (green line) coincides with one of the bare J-C absorption peaks. The other parameters used for all the calculations in (A–D) are ħγb=2 meV, ħγσ=2×10−5 eV, ħγa=150 meV, ħg=100 meV, ħΔ=ħδ=0 eV.

The dephasing-induced broadening also affects the J-C emission spectra sE(ω) shown in Figure 5B for ħℰ=1 meV and ħΔ=0 eV. When the dephasing is switched on, there appears a strong inelastic contribution to the light emission originating predominantly from the lower polariton branch, as the upper branch is efficiently depopulated by the incoherent energy transfer. Without dephasing, the emission spectrum practically vanishes as the incoherent population of the polaritons is low.

Finally, we turn on the interaction between the molecular electronic and vibrational levels and calculate the effect of dephasing on the emission spectra of the complete system. The resulting emission spectra are shown in Figure 5C for the same parameters as the emission spectra of the bare polaritons in Figure 5B and for d=0.1, ħωm=50 meV and ħg=100 meV. All of the spectra feature a sharp Stokes–Raman peak at the frequency ωLωm, which for ħγϕ=10 meV and 50 meV is accompanied by the broad and intense fluorescence background already discussed in Figure 5B. Particularly for ħγϕ=50 meV, the background overwhelms the relatively weak SERS signal. For no dephasing, this background is strongly reduced and becomes negligible in comparison with the Raman peak.

For completeness, we show in Figure 5D the incoherent population of the vibrational level for the same parameter d=0.1, pumping amplitude ħℰ=1 meV, and for ħγϕ=10 meV. The results as a function of laser detuning and vibrational frequency can be compared with those in Figure 3A, where we plotted the same results except that we adopted ħγϕ=0 eV there. As expected, the most noticeable feature introduced by dephasing is the larger width of the displayed features and the lower populations.

## 7 Effect of strong illumination

We now consider the situation where the intensity of the incident laser is strong, while the electron–vibration coupling is assumed to be weak, and in the absence of pure dephasing. An intense laser populates the different manifolds of the J-C ladder and thus leads to nonlinear effects in the resonant SERRS signal that were not considered in our weak-illumination expressions. A full analytical treatment of this situation is very challenging, but we can still gain insight into the vibrational dynamics, assuming the same separation of time scales as discussed before, by inserting the numerically calculated reservoir spectral function J(s) into the expression of the decay and pumping rates (11 and 12) and then using these values in the rate equations of the vibrations (9 and 10) to obtain the steady-state value of the vibrational population (14). We complement this simplified semianalytical approach by full numerical calculations. For simplicity, we continue using the rotation wave approximation, although for the strongest laser considered, a correction may be necessary.

We plot in Figure 6 the incoherent vibrational population as a function of the amplitude ℰ and detuning Δ of the illuminating laser, obtained using the full-numerical (Figure 6A) and the semianalytical (Figure 6B) approach, and for d=0.1, ħωm=50 meV and ħg=100 meV. The excellent agreement between the two supports the assumption that the use of very intense laser illumination does not invalidate the separation of the system into the slowly evolving vibrations and the fast J-C reservoir.

Figure 6:

Nonlinear resonant surface-enhanced Raman scattering (SERS) response for strong laser illumination.

(A, B) Incoherent population of a vibrational mode of energy ħωm=50 meV as a function of the laser detuning and pumping amplitude ℰ calculated from (A) the full numerical model and (B) the simplified model. The vibrations are weakly coupled to the electronic states (d=0.1), and there is no pure dephasing. (C, D) Evolution of the SERS emission with pumping amplitude (ℰ), calculated numerically for a single vibrational mode of energy ħωm=50 meV that is weakly coupled to the electronic states (d=0.1) (C) Emission spectra as a function of ℰ for detuning of the incident laser ħΔ=0 eV. (D) Maximum intensity of the Stokes (blue line) and anti-Stokes (green line) Raman line as a function of ℰ [shown in (C) for up to ħℰ=48 meV]. To obtain the value of the Raman peaks, we first subtracted the background, defined as the inelastic emission of the J-C system. The other parameters used for all calculations are as follows: ħγb=2 meV, ħγσ=2×10−5 eV, ħγa=150 meV, ħg=100 meV, ħγϕ=0 eV, Δ=δ.

More strikingly, the incoherent population is strongly nonlinear with respect to the intensity, ∝|ℰ|2, of the incident laser. For weak illumination, we are in the linear regime, where the incoherent population shows local maxima for two different moderate detunings, as obtained from (14), together with (11), (12), and (15), analogously to the results in the previous section. The incoherent population of these peaks initially grows when increasing ℰ as more energy is pumped into the system, but for a certain laser intensity, the incoherent population reaches a maximum and starts to decay. We attribute this behaviour to the underlying nonlinearity of the J-C system due to the uneven spacing of the energy levels sketched in Figure 1B.

An intense illumination can also affect very strongly the SERRS spectra measured in an experiment. We study this situation with use of the full-numerical calculations and, as an example, we show the inelastic emission spectra as a function of pumping amplitude ℰ for zero detuning Δ=0 (Figure 6C, notice the logarithmic scale). The calculated emission spectra can still be regarded as a combination of the resonance fluorescence background and the Raman lines. For low illumination intensities, the spectra in Figure 6C feature a clear Raman–Stokes peak at emission frequency ω=ωLωm with a strength that is significantly stronger than that of the background. One can also distinguish a second-order Stokes peak at ω=ωL–2ωm and the anti-Stokes peak at ω=ωL+ωm. As the intensity of the laser is increased, but still sufficiently small, the strength of both the background and Raman peaks increases, with the latter remaining clearly visible. In contrast, for very large laser intensities ħℰ⪆100 meV, the amplitude of the Raman peaks is reduced, and the spectrum is fully dominated by the background.

The laser intensity also affects the spectral shape of the background. For example, the two broad background peaks in Figure 6C for ħℰ=1 meV become a single broad flat feature for ħℰ=48 meV. Therefore, the laser intensity does not simply scale the emission spectra by a frequency-independent factor, but it changes the overall spectral shape and affects the relative weights of the Raman lines and the background.

Finally, we highlight in Figure 6D the nonlinear dependence of the maximum of the SERS signal on the incident laser amplitude by plotting the background-free maxima of the Raman–Stokes (blue line) and anti-Stokes (green line) lines. The background that we subtract from the total signal corresponds to the inelastic emission spectrum of the J-C system uncoupled from the vibrational modes. The results show that the intensity dependence of the Stokes and anti-Stokes amplitude resembles the result for the incoherent vibrational populations (Figure 6A and B), with a broad maximum that peaks at an optimal pumping amplitude ℰ. Interestingly, the optimal illumination-field amplitude is different for the Stokes and the anti-Stokes lines. The maximum of the anti-Stokes line appears at roughly the pumping amplitude ħℰ≈70 meV, while the Stokes emission peaks are maximised at lower powers ħℰ≈45 meV.

We notice that such illumination strengths correspond to very intense lasers that can destroy the sample [the sample damage threshold estimated in [5] W=102 μW/μm2 (W∝ℰ2), can be roughly expressed in terms of the pumping amplitude as ħℰ≈10 meV] and may require pulsed illumination [19]. Further, for such strong lasers, a more rigorous calculation will require going beyond the rotating-wave approximation used in this article, and other effects (anharmoniticies of the vibrations, excitation of other electronic states) can become important.

## 8 Outlook and conclusions

We have studied the resonant SERS process for a system where the molecular exciton is strongly coupled to a single plasmonic mode, forming thus a set of polaritons that interact with a vibrational mode of the same molecule. To bring insight into the underlying physical mechanisms behind this inelastic process in this system, we have first developed an intuitive analytical description of the vibrational pumping and associated Raman scattering for weak-illumination conditions ℰ, weak electron–phonon coupling, and no pure dephasing. The analytical results match the full numerical calculations very well and allow for a simple interpretation of the vibrational pumping mechanism as a two-step process that is similar to the mechanism discussed for nonresonant SERS [16], [17], [18], [58] and SERRS in the weak-coupling regime [28] but for a more complex reservoir. In the first step, the amplitude of incident light excites a coherent population of the electronic state in the molecule. In a second step, the Raman–Stokes excitation is spontaneously transferred into the J-C system formed by the coupling of the plasmon and the electronic TLS. This excitation is then either absorbed or re-emitted into the far field.

We have also studied more complex situations. We first considered the influence of two intrinsic parameters of the system, namely, the electron–phonon coupling, S=d2, and the pure dephasing, γϕ. Larger values of d promote higher-order vibrational transitions and generally increase the interaction strength between the electronic and vibrational states of the molecule, leading to the breakdown of the simple reservoir-system picture introduced for small d and to the emergence of higher-order Raman peaks in the emission spectra. On the other hand, pure dephasing reduces the efficiency of vibrational pumping, leading to smaller final incoherent vibrational populations, and thus diminishes the intensity of vibrational Raman peaks in the inelastic emission spectra. Dephasing also significantly broadens and enhances the fluorescence background. As a consequence of the weaker lines and stronger background, the visibility of the Raman lines becomes considerably smaller, which complicates the realisation of SERRS experiments.

Last, we have addressed the effect of increasing the intensity of the illumination. We have shown that the interpretation of the SERRS process as a two-step enhancement process as described above remains valid, but the system becomes strongly nonlinear due to the nonhomogeneous separation of the polaritonic levels of the J-C system (J-C ladder). The strong nonlinear response of the two-level transition likely stands behind the counterintuitive finding of a maximum of the incoherent vibrational population and SERRS signal for a particular value of the pumping laser amplitude.

## Acknowledgements

The authors acknowledge project FIS2016-80174-P from the Spanish Ministry of Education, H2020-FETOPEN project “THOR” Nr. 829067 from the European Commission, grant IT1164-19 for consolidated groups of the University from the Basque Government, and project PI2017-30 of the Departamento de Educación, Política Lingüística y Cultura of the Basque Government.

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## Article note

The data used in this paper can be found in http://hdl.handle.net/10261/196160.

## Footnotes

• 1

Throughout the article, we use the term absorption even though (4) strictly speaking defines the extinction spectrum (i.e. defines the energy lost by the weak probe due to the interaction with the plasmon–exciton system).

Revised: 2019-10-22

Accepted: 2019-10-26

Published Online: 2019-12-20

Published in Print: 2020-02-25

Citation Information: Nanophotonics, Volume 9, Issue 2, Pages 295–308, ISSN (Online) 2192-8614,

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Here we are providing Class 11 chemistry Important Extra Questions and Answers Chapter 5 States of Matter. Chemistry Class 11 Important Questions are the best resource for students which helps in Class 11 board exams.

## Class 11 Chemistry Chapter 5 Important Extra Questions States of Matter

### States of Matter Important Extra Questions Very Short Answer Type

Question 1.
What is the pressure of a gas? What is its S.I. unit?
The force exerted by the gas molecules per unit area on the walls of the container is equal to its pressure SI. unit of pressure is the pascal (Pa).
1 Pa = Nm2- =1 kg m-1s-2.

Question 2.
What does the temperature of gas represent?
The temperature of the gas represents the average kinetic energy of the gas molecules as we know K.E. ∝ $$\sqrt{T}$$.

Question 3.
Why is it not possible to cool gas to o K?
This is because all gases condense to liquids or solids before this temperature is reached.

Question 4.
What property of molecules of real gases is indicated by van der Waal’s constant ‘a’?
Intermolecular attraction.

Question 5.
What do you understand by standard temperature?
The standard temperature is 0°C or 273 K.

Question 6.
What is the effect of temperature on the vapour pressure of a liquid?
Vapour pressure increases with rising in temperature.

Question 7.
What is an ideal gas equation?
For given ideal gas PV = nRT
where P = gas pressure;
V = volume,
R = gas constant
T = Temp. (Kelvin scale);
n – no. of moles of gas.

Combined Gas Law Calculator … Combined gas law is the combination of Charles’s law, Boyle’s law, and Gay-Lussac’s law.

Question 8.
Which state of matter has a definite volume, but no definite shape?
Liquid.

Question 9.
What is the value of the gas constant in S.I. units?
8.314 JK-1 mol-1.

Question 10.
What are the S.I. units of surface tension?
Nm-1

Question 11.
What is the compressibility factor?
Z = $$\frac{PV}{nRT}$$.

Question 12.
What is the equation of state for real gases?
van der Waal’s equation(P + $$\frac{a}{\mathrm{~V}^{2}}$$)(v – b) = RT for 1 mole.

Question 13.
How is the pressure of a gas related to its density at a particular temperature?
d = $$\frac{MP}{RT}$$

Question 14.
How is the partial pressure of a gas in a mixture related to the total pressure of the gaseous mixture?
The partial pressure of a gas = Mole fraction of that gas × total pressure.

Question 15.
What is the relationship between average kinetic energy , and the temperature of a gas?
K.E. = $$\frac{3}{2}$$kT
where k is Boltzmann constant = $$\frac{\mathrm{R}}{\mathrm{N}_{0}}$$.

Question 16.
What is the significance of van der Waal’s constant ‘a’ and ‘b’?
‘a’ is a measure of the magnitude of the intermolecular forces of attraction while ‘b’ is a measure of the effective size of the as molecules.

Question 17.
Arrange the solid, liquid and gas in order of energy-giving reasons.
Solids < liquid < gas. This is because a solid absorbs energy to change into a liquid which further absorbs energy to change into a gas.

Question 18.
What is the effect of pressure on the boiling point of a liquid?
The boiling point increases as the prevailing pressure increases.

Question 19.
Why are vegetables cooked with difficulty at a hill station?
The atmospheric pressure decreases as we go up. Therefore at the hills due to the lowering of atmospheric pressure, boiling points lowered.

Question 20.
The size of the weather balloon keeps on becoming larger as it rises to high altitude. Explain why?
At higher altitudes, the external pressure on the balloon decreases and therefore, its size increases.

Question 21.
How is the pressure of a given sample of a gas related to the temperature at a constant volume?
P ∝ T, i.e., Pressure varies as absolute temperature.

Question 22.
How is the pressure of a gas related to the number of molecules of a gas at constant volume and temperature?
P ∝ N; N = No. of molecules of a gas.

Question 23.
What type of graph do we get when we plot a graph PV against P? What is shown by this graph?
It is a straight line parallel to the r-axis. It shows that PV is constant at different pressures. It shows Boyle’s Law.

Question 24.
At what temperature will oxygen molecules have the same kinetic energy as ozone molecules at 30°C?
AT 30°C. Kinetic energy depends only on the absolute temperature and not on the identity of the gas.

Question 25.
At a particular temperature why vapours pressure of acetone is less than that of ether?
This is because intermolecular forces in acetone are more than those present in ether.

Question 26.
Out of NH3 and N2 which will have
(i) a larger value of V and
NH3 will have a large value of ‘a’ because of hydrogen bonding.

(ii) larger value of ‘b’?
N2 should have a large value of ‘b’ because of its larger molecular size.

Question 27.
Why are the gases helium and hydrogen not liquefied at room temperature by applying very high pressure?
Because their critical temperature is lower than room temperature. Gases cannot be liquefied above the critical temperature by applying even very high pressure.

Question 28.
What will boil at a higher temperature at sea level or at the top of mountains?
Water will boil at a higher temperature at sea level.

Question 29.
Under what conditions do real gases tend to show ideal gas behaviour?
When the pressure of the gas is very low and the temperature is high.

Question 30.
Both propane (C3Hg) and carbon dioxide (CO2) diffuse at the same rate under identical conditions of temperature and pressure. Why?
Both propane (C3Hg) and carbon dioxide (CO2) have the same molar mass (44 gm).

Question 31.
If the number of moles of a gas is doubled by keeping the temperature and pressure constant, what will happen to the volume?
The volume will also double as V ∝ n according to Avogadro’s Law.

Question 32.
What is a Triple point?
The temperature at which solid, liquid and vapour; i.e., all the three states of the substance exist together is called the triple point.

Question 33.
Is Dalton’s law of partial pressures valid for a mixture of SO2 and O2?
No, the law holds good only for those gases which do not react with each other.

Question 34.
Under what conditions a gas deviates from ideal gas behaviour?
It deviates at low temperature and high pressure.

Question 35.
Molecule A is twice as heavy as molecule B. Which of these has higher kinetic energy at any temperature?
K.E. of a molecule is directly proportional to temperature and is independent of its mass. So both the molecule A and B at any temperature will have equal kinetic energy.

Question 36.
How will you define pascal?
It is defined as the pressure exerted by a force of one newton on an area of one meter2.
Pa = 1 Nm-2.

Question 37.
How will you define London or dispersion forces?
The forces of attraction between the induced momentary dipoles are called London or dispersion forces.

Question 38.
What is Absolute Zero?
The lowest possible hypothetical temperature of – 273°C at which gas is supposed to have zero volume is called Absolute zero.

Question 39.
What is the nature of the gas constant R?

= work done per degree per mole.

Question 40.
What is the value of R in SI units?
R = 8.314 JK-1 mol-1 = 8.314 k Pa dm3 K-1 mol-1

Question 41.
What is meant from Boyle point or Boyle temperature?
The temperature at which a real gas behaves like an ideal gas over an appreciable pressure range is called Boyle point or Boyle temperature.

Question 42.
How will you define viscosity?
The internal resistance to the flow of a fluid is called its viscosity.

Question 43.
What is the effect of temperature on the viscosity of a liquid?
The viscosity of a liquid decreases with an increase in temperature.

Question 44.
How will you convert pressure in atmospheres into SI units?
1 atm = 1,01,325 Pa or Nm-2 or 1 bar = 105 Pa.

Question 45.
What do you understand from surface tension?
The surface tension of a liquid is defined as the force acting at right angles to the surface along a one-centimetre length of the. liquid.

Question 46.
What is the effect of the increase in temperature on the surface tension of a liquid?
Surface tension decreases in increasing the temperature.

Question 47.
What is the difference between normal boiling point and standard boiling point?
When the external pressure is equal to one atmospheric pressure the boiling point is referred to as a normal boiling point, when it is one bar, the boiling point is called its standard boiling point.

Question 48.
What is the difference between vapour and gas?
When gas is below its critical temperature, it is called vapour.

Question 49.
What happens if a liquid is heated to the critical temperature of its vapours?
The meniscus between the liquid and the vapour disappears, the surface tension of the liquid becomes zero.

Question 50.
Why falling liquid drops are spherical?
The surface area of a sphere is minimum. In order to have minimum surface area drops of the liquid become spherical.

### States of Matter Important Extra Questions Short Answer Type

Question 1.
Ammonia and Sulphur dioxide gases are prepared in two comers of a laboratory. Which gas will be detected first by a student working in the middle of the laboratory and why?
Molecular mass of NH3 = 17 Molecular mass of SO2 = 64
NH3 is a lighter gas and diffuses at a faster speed than SO2
∴ NH3 gas will be detected first.
[∵ Acc. to Graham’s law of diffusion $$\frac{r_{1}}{r_{2}}=\sqrt{\frac{d_{2}}{d_{1}}}$$

Question 2.
What is the effect of hydrogen bonding on the viscosity of a liquid?
Hydrogen bonding leads to an increase in the effective size of the moving unit in the liquid. Due to an increase in the size and mass of the molecule, there is greater interval resistance of the flow of the liquid. As a result, the viscosity of the liquid rises.

Question 3.
Which are the two faulty assumptions in the kinetic theory of gases.

1. There is no force of attraction between the molecules of the gas.
2. The volume of the molecules of the gas is negligibly small as compared to the total space (volume) occupied by the gas.

Question 4.
What is the relationship between the density and molar mass of a gaseous substance? Derive it.
Ideal gas equation is PV = nRT
or $$\frac{n}{V}=\frac{p}{R T}$$
Replacing n by $$\frac{m}{M}$$, we get

Question 5.
What is meant by the term: Non-ideal or real gas?
The gas which does not obey the Cas law:
Boyle’s law, Charles’ law, Avogadro^ law at all temperatures and pressures is a called-non-ideal or real gas. Most of the real gases show ideal behaviour at low pressure and high temperature.

Question 6.
Derive the ideal gas equation PV = nRT.
According to Boyle’s law V ∝ $$\frac{1}{P}$$ if n and T are constant.
According to Charles’ law V ∝ T at constant P.and n
According to Avogadro’s law V ∝ n at constant T and P
Combining the three laws
In
V ∝ $$\frac{Tn}{P}$$
or
PV ∝ nT
or
PV = nRT where R is a constant of proportionality called universal gas constant.

Question 7.
Why liquids have a definite volume, but no definite shape?
It is due to the fact that in liquids intermolecular forces are strong enough to hold the molecules together, but these are strong enough to hold the molecules together, but these forces are not strong enough to fix them into definite or concrete positions as in solids. Hence they possess fluidity but no definite shape.

Question 8.
How do the real gases deviate from ideality above and below the Boyle point?
Above their Boyle point, real gases show positive deviations from ideality and the values of Z are greater than one. The forces of attraction between the molecules are very feeble. Below the Boyle point, real gases first show a decrease, in Z value with increasing pressure, the value Of Z increases continuously.

Question 9.
Write down the van der Waals, equation for n moles of a real gas. What do the constants ‘a’ and ‘b’ stand for?
(P + $$\frac{a n^{2}}{V^{2}}$$) (V – nb) = nRT
where p, Vindicate gas pressure and its volume. T is the Kelvin temperature of the gas. R is gas constant. Value of ‘a’ is a measure of the magnitude of intermolecular attractive forces within the molecule and is independent of temperature and pressure, ‘rib’ is approximately the total volume occupied by the molecules themselves. ‘a’ and ‘b’ are called van der Waals constants and their value depends upon the nature of the gas.

Question 10.
How is compressibility factor Z related to the real and ideal volume of the gas?
Z = $$\frac{P \mathrm{~V}_{\text {real }}}{n \mathrm{RT}}$$ …(1)

If the gas shows ideal behaviour, then
Videal = $$\frac{nRT}{P}$$ ….(2)

On putting the titis value of nRT in equation (1), we get
Z = $$\frac{\mathrm{V}_{\text {real }}}{\mathrm{V}_{\text {ideal }}}$$

Thus the compressibility factor Z is the ratio of the actual molar volume of a gas to the molar volume of it. if it were an ideal gas at that temperature and pressure.

Question 11.
What do the critical temperature, critical pressure, and critical volume for a gas stand for?
The critical temperature is the temperature above which a gas cannot be liquified however large may be the pressure applied on it. The pressure sufficient to liquiíy a gas at its critical temperature is called its critical pressure. The volume of the gas at its crìtical temperature is called its critical volume’.

Question 12.
What do the absolute zero and absolute scale of temperature stand for?
The lowest possible hypothetical temperature of – 273°C at which gas is supposed to have zero volume is called Absolute! zero.

Lord Kelvin suggested a new scale of temperature starting with – 273°C and it’s zero. This scale of temperature is called the absolute scale of temperature.

Absolute or Kelvin temperature is given by T, where
T = t°C + 273
t°C stands for the centigrade scale of temperature.

Question 13.
Give one application of Dalton’s law of partial pressures.
One application of Dalton’s law of partial pressures is in determining the/pressure of dry gas. Gases are generally collected over water and, so they contain water vapours. The pressure exerted by the water Vapours at-a particular temperature is called the Aqueous Tension. By subtracting the aqueous tension from the ‘ vapour pressure of the moist gas, the pressure of the dry gas can be calculated.
Pdry gas= Pmoist gas – Aqueous Tension (at t°C).

Question 14.
How will you calculate the partial pressure of gas using Dalton’s Law of partial pressures?
In a mixture of non-reacting gases A, B, C etc., if each gas is considered to be an ideal gas, then
PA = nA $$\frac{RT}{V}$$
PB = nB $$\frac{RT}{V}$$
PC = nC $$\frac{RT}{V}$$
where nA, nB nC stand for their respective moles in the same vessel .
[V = constant, keeping T constant]
Then according to Dalton’s law ‘

Thus, particle pressure of a. gas – Mole fraction of A × Total pressure.

Question 15.
Is there any effect of the nature of a liquid and temperature on the surface tension of a liquid?

1. Effect of nature of the liquid: Surface tension is a property that arises due to the intermolecular forces of attraction among the molecules of the liquid.
2. The surface tension of the liquids generally decreases with an increase in temperature and becomes zero at the critical temperature.

Question 16.
Leaving electrostatic forces that exist between oppositely charged ions, which other intermolecular attractive forces exist between atoms or ions. Mention them.

1. van der Waals forces
2. London Forces or Dispersion Forces
3. Dipole-dipole forces
4. Dipole-induced dipole forces
5. Hydrogen bond.

Question 17.
What is the relationship between intermolecular forces and thermal interactions?
Intermolecular forces tend to keep the molecules together but the thermal energy of the molecules tends to keep them apart. Three states of matter are the result of a balance between intermolecular forces and the thermal energy of the molecules.
Gas → Liquid → Solid
intermolecular forces increases →
Gas ← Liquid ← Solid
increase in Thermal energy ←

Question 18.
What are the main characteristics associated with gases?

1. Gases are highly compressible.
2. Gases exert pressure equally in all directions.
3. Gases have a much lower density than solids or liquids.
4. Gases do not have definite shape and volume.
5. Gases intermix freely and completely in all proportions.

Question 19.
What do you understand by the term? Isotherms and Isobars?
Isotherms are the curves plotted by varying pressure against volume keeping temperature constant.

Each line obtained by plotting volume against temperature keeping, pressure constant is called an Isobar.

Question 20.
What are the forces which are responsible for the viscosity of a liquid? Why is glycerol more viscous than water?
The forces responsible for the viscosity of the liquids are

• Hydrogen bonding
• van der Waals torches.

Glycerol possesses greater viscosity than water because glycerol has extensive hydrogen bonding in its molecules due to the pressure of three-Oil groups in it as compared to the hydrogen bonding in water molecules.

### States of Matter Important Extra Questions Long Answer Type

Question 1.
Derive the Ideal gas equation PV = nRT.
Let a certain mass of a gas at pressure P1 and temperature
T1 occupy a volume V1. On changing the pressure and temperature respectively to P2 and T2, let the volume of the gas change to V2.
(i) Let us first change the pressure P1 to P2 at constant temperature T1. Then, according to Boyle’s law, volume V is given by,
P2V = P1V1
or
V = $$\frac{P_{1} V_{1}}{P_{2}}$$ …(1)

(ii) Now keeping the pressure constant at P2, heat the gas from temperature T1 to T2, the volume changes from V to V2.
By Charle’s Law, we must have,

The numerical value of k depends upon the amount of gas and the units in which P and V are expressed. By Avogadro’s law, for one mole of any gas, the value of $$\frac{PV}{T}$$ will be the same and is represented
$$\frac{PV}{T}$$ = R
or
PV = RT.
for n moles of an ideal gas
PV = nRT

Question 2.
State the fundamental assumptions of the kinetic theory of gases
Or
What are the postulates of the kinetic theory of gases?
Or
Write a short note on “Microscopic Model of a gas”.
To explain the general properties of gases to provide some theoretical explanation for various gas laws (stated on basis of experimental studies), various, scientists Bernoulli, Clausius, Maxwells, Boltzmann and others, gave ‘Kinetic Theory of Gases’ which explains qualitatively as well as quantitatively the various aspects of gas behaviour.

The important postulates of the theory are:

1. All gases are made up of a very large number of minutes particles called molecules.
2. The molecules are separated from one another by large distances. The empty spaces among the molecules are so large that the actual volume of the molecules is negligible as compared to the total volume of the gas.
3. The molecules are in a state of constant rapid motion in all directions. During their motion, they keep on colliding with one another and. also with the walls of the container and, thus, change their directions.
4. Molecular collisions are perfectly elastic, no loss of energy occurs when the molecules collide with one another or with the walls of the container. However, there may be. redistribution of energy during the collisions.
5. There are no forces of interaction (attractive or repulsive) between molecules. They move completely independent of one another.
6. The pressure exerted by the gas is due to the bombardment of its molecules on the walls of the container per unit area.
7. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature.

Question 3.
Explain the following observations.
(a) Aerated water bottles are kept underwater during summer.
Aerated water contains CO2 gas dissolved in an aqueous solution under pressure and the bottles are well stoppered. As in summer temperature increases and we know that the solubility of the gases decreases with increase in temperature and as a result move of gas is expected to be generated may be large in quantity and hence pressure exerted by the gas may be very high and the bottle may explode. So, to decrease the temperature and hence to avoid explosion of the bottles.

(b) Liquid ammonia bottle is cooled before opening the seal.
Liquid ammonia bottle contains the gas under very high pressure. If the bottle is opened as such, then the sudden decrease in pressure will lead to a large increase in the volume of the gas. As a result, the gas will come out of the bottle with force. This will lead to the breakage of the bottle. Cooling under tap water will result in a decrease in volume. It reduces the chances of an accident.

(c) The tyre of an automobile is inflated at lesser pressure in summer than in winter.
The pressure of the air is directly proportional to the temperature. During summer due to high temperature, the pressure in the tyre will be high as compared to that in water. The tube may burst under high pressure in summer. Therefore, it is advisable to inflate the tyre to lesser pressure in summer than in winter.

(d) The size of the weather balloon becomes larger and larger as it ascends up to higher altitudes.
Answered somewhere else in the book.

### States of Matter Important Extra Questions Numerical Problems

Question 1.
In a hospital, an oxygen cylinder holds 10 L of oxygen at 200 atm pressure. If a patient breathes in 0.50 ml of oxygen at 1.0 atm with each breath, for how many breaths the cylinder will be sufficient. Assume that all the data is at 37°C.
10 L at 200 atm =? L at 1 atm. [Temp, is constant at 37°C]
∴ Boyle’s law P1V1 = P2V2 can be applied
200 × 10 = 1 × V2
or
V2 = 2000 L

Number of breaths = $$\frac{\text { Total volume }}{\text { Volume inhaled per breath }}$$
= $$\frac{2000 \mathrm{~L}}{0.5 \times 10^{-3} \mathrm{~L}}$$
= 4 × 106

Question 2.
Calculate the pressure of 1 × 1022 molecules of oxygen gas when enclosed ¡n a vessel of 5-litre capacity at a temperature of 27°C.
Number f moles of O2 = $$\frac{1 \times 10^{22}}{6.02 \times 10^{23}}$$

= 0.166 × 10-1 mol
= 1.66 × 102 mol
The pressure exerted by O2 molecules can be calculated by the equation
PV = nRT [Ideal gas equation]

Question 3.
When a ship is sailing in the Pacific Ocean where the temperature is 23.4°C, a balloon is filled with 2 L air. What will be the volume of the balloon when the ship reaches the Indian Ocean, where the temperature is 26.1°C.
V1 = 2 L
T1 = 23.4 + 273 = 296.4K
T2 = 26.1 + 273 = 299.1 K
From Charle’s Law

Question 4.
At what temperature centigrade, will the volume of a gas at 0°C double itself, pressure remaining constant?
Let the volume of the gas at 0°C = V1 mL
Final Volume = V2 = 2V1 mL [Given]
T1 = OC + 273 = 273 K, T2 =?

Since Pressure remains constant
∴ Charles law can be applied

Question 5.
A bulb ‘B’ of the unknown volume containing gas at one atmospheric pressure is connected to an evacuated bulb of 0.5-litre capacity through a stop-cock. On opening the stop-cock, the final pressure was found to be 570 mm at the same temperature. ‘What is the volume of bulb ‘B’?
Let the volume of the bulb = V1; P, = 1 atm
Evacuated Bulb has volume = 0.5 L;
Total volume of both bulbs after the opening of stop-cock = (V1 + 0.5) L i.e. V2 = (V1 + 0.5):
P2 = 570 mm = $$\frac{570}{760}$$ atm

Apply Boyle’s law P1V1 = P2V2
1 × V1 = $$\frac{570}{760}$$ × (V1 + 0.5)

On solving the above equation
V1 = 1.5 L
∴ Volume of the unknown bulb ‘B’ = 1.5 L

Question 6.
1 mole of SO2 occupies 1.5 L at 25°C. Calculate the pressure exerted by gas assuming that gas does not obey the ideal gas equation. (Given a = 3.6 atm L2mol-2, b = 0.04 L mol-1) where a, b are van der Waal’s constants.

Question 7.
A gas occupies a volume of 2.5 L at 9 × 105 Nm-2. Calculate the additional pressure required to decrease the volume of the gas to 1.5 L, keeping the temperature constant.
V1 = 2.5 L; P1 = 9 × 105 Nm-2
V2 = 1.5 L; P2 =?

Since temperature is constant, Boyle’s law is applied.

Question 8.
What volume of air will be expelled from a vessel containing 400 cm3 at 7° C when it is heated to 27° C at the same pressure?
As the pressure is constant, Charles law is applied

428.6 cm3 is the volume after expansion
∴ Volume expelled = (428.6 – 400) cm3 = 28.6 cm3.

Question 9.
A 10.0 L container is filled with a gas to a pressure of 2.0 atm at 0°C. At what temperature will the pressure of the gas inside the container be 2.50 atm.
As the volume of the container remains constant, Gay- Lussac’s Law/Amonton’s Law is applicable
$$\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$$
$$\frac{2}{273}=\frac{2.50}{\mathrm{~T}_{2}}$$
T2 = 341 K
or
t2 = (341 – 273)°C = 68°C.

Question 10.
One litre flask containing vapours of methyl alcohol (Mol. mass = 32) at a pressure of 1 atm and 25°C was evacuated till the final pressure was 10-3 mm. How many molecules of methyl alcohol were left in the flask?
P1 = 10-3 mm; V1 = 1000 cm3; T1 = 298 K
Converting this volume to volume, at STP, where T2 = 2/3 K and P2 = 760 mm

Now, 22400 cm3 at STP contains 6.02 × 1023 molecules

∴ 1.205 × 10-3 cm3 at STP contains
= $$\frac{6.02 \times 10^{23}}{22400}$$ × 1.205 × 10-3 molecules
= 3.24 × 1016 molecules.

Question 11.
A sealed tube that can withstand a pressure of 3 atm is filled with air at 27°C and 760 mm pressure. Find the temperature above which it will burst?
Applying gas equation $$\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$$ as volume of the sealed tube remains constant .
$$\frac{1}{300}=\frac{3}{\mathrm{~T}_{2}}$$
T2 = 900 K
or
t2 = 900 – 273 = 627°C.

Question 12.
The temperature at the foot of a mountain is 30°C and pressure is 760 mm. Whereas at the top of the mountain these are 0°C and 710 mm. Compare the densities of the air at the foot and at the top.

∴ the ratio of densities of air at the foot and the top of the mountain = 0.964: 1.

Question 13.
10.0 g of O2 were introduced into an evacuated vessel of 5-litre capacity maintained at 27°C. Calculate the pressure of the gas in the atmosphere in the container.
Volume of the gas = V = 5 L;
Wt. of O2 = 10.0 g
Molar mass of O2 = 32.0
∴ No. of moles = $$\frac{10}{32}$$

T = 27 + 273 = 300 K; R = 0.0821 L atm K-1 mol-1
From ideal gas equation,
PV = nRT, we get
P = $$\frac{10}{32}$$ × 0.0821 × 300 = 1.54 atm.

Question 14.
8.0 g of methane is placed in a 5 L container at 27°C. Find Boyle constant.
Pressure × Volume is called Boyle’s constant
PV = nRT = $$\frac{W}{M}$$ RT ,
= $$\frac{8}{16}$$ × 0.0821 × 300
= 12.315 L atm

Question 15.
The density of A gas is 3.80 g L-1 at STP. Calculate its density at 27°C and 700 torr pressure
d = $$\frac{PM}{RT}$$
For the same gas at two different pressures and temperatures,

Question 16.
If the density of a gas at sea level and 0°C is 1.29 kg m-3, what will be its molar mass (Assume that pressure is equal to 1 bar).

Question 17.
A gaseous mixture contains 56 g N2, 44 g CO2 and 16 g CH4 The total pressure of the mixture is 720 mm Hg. What is the partial pressure of CH4?

Question 18.
A 2 L flask contains 1.6 g of methane and 0.5 g of hydrogen at 27°C. Calculate the partial pressure of each gas in the mixture and hence calculate the total pressure.
1.6 g CH4 = $$\frac{1.6}{16}$$ = 0.1 Mole
16
Partial pressure of CH4 (pCH4) = CH4 × $$\frac{RT}{P}$$

Question 19.
One mole of S02 gas occupied a volume of 350 mL at 27°C and 50 atm pressure. Calculate the compressibility factor of the gas. Comment on the type of deviation shown by the gas from ideal behaviour.
Compressibility factor, Z =
Here n = 1, P = 50 atm, V = 350 × 10-3 L = 0.35 L
R = 0.0821 L atm K-1 mol-1; T = 27 + 273 = 300 K
∴ Z = $$\frac{50 \times 0.35}{1 \times 0.0821 \times 300}$$ = 0.711
For an ideal gas Z = 1
As for the given gas Z < 1, it shows a negative deviation, i.e., it is more compressible than expected from ideal behaviour.

Question 20.
A mixture of CO and CO2 is found to have a density of 1.5 g L-1 at 20°C and 740 mm pressure. Calculate the composition of the mixture.
1. Calculation of average molecular mass of the mixture
M = $$\frac{d R T}{P}=\frac{1.50 \times 0.0821 \times 293}{\frac{740}{760}}$$ = 37.06

2. Calculation of percentage composition
Let mol% of CO in the mixture = x
Then mol% of CO? in the mixture = 100 – x

Average molecular mass

Question 21.
A 5-L vessel contains 14 g of nitrogen. When heated to 1800 K, 30% of molecules are dissociated into atoms. Calculate the pressure of the gas at 1800 K.
N2 ⇌ 2N

Initial moles = $$\frac{1.4}{28}$$ = 0.05
Moles left = 0.05 – $$\frac{30}{100}$$ × 0.05 = 2 × 0.015 = 0.03

∴ Total no. of moles = 0.035 + 0.030 = 0.065
i.e. n = 0.065 mol, V = 5 L, T = 1800 K; P =?
P = $$\frac{n \mathrm{RT}}{\mathrm{V}}=\frac{0.065 \times 0.0821 \times 1800}{5}$$
= 1.92 atm.

Question 22.
Calculate the temperature of 2 moles of S02 gas contained in a 5 L vessel at 10 bar pressure. Given that for SO2 gas van der Waal’s constant is a – 6.7 bar L2 mol-2 and b = 0.0564 L mol-1.
According to van der Waal’s equation

Question 23.
A given mass of a gas occupies 919.0 mL in a dry state at STP. The same mass when collected over water at 15°C and 750 mm pressure occupies on litre volume. Calculate the vapour pressure of water at 15°C.
If p is the vapour pressure of water at 15°C, then P2 = 750 – p
From the gas equation $$\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}$$, we get
$$\frac{760 \times 919}{273}=\frac{(750-p) \times 1000}{288}$$
or
p = 13.3 mm

∴ Vapour pr. of water = 13.3 mm.

Question 24.
A steel tank containing air at 15 atm pressure at 15°C ¡s provided with a safety valve that will yield at a pressure of 30 atm. To what minimum temperature must the air be heated to below the safety valve?
$$\frac{P_{1}}{P_{2}}=\frac{T_{1}}{T_{2}}$$
i.e., $$\frac{15}{30}=\frac{288}{\mathrm{~T}_{2}}$$
or
T2 = 576 K
or
t2°C = 576 – 273 = 303°C

Question 25.
Calculate the pressure exerted by 110 g of CO2 in a vessel of 2 L capacity at 37°C. Given that the van der Waals constants are a = 3.59 L2 atm mol-2 and b = 0.0427 L mol-1. Compare the value with the calculated value if the gas were considered ideal.
Here, n = $$\frac{110}{44}$$ = 2.5 moles. Putting the given values, we get
P = $$\frac{2.5 \times 0.0821 \times 310}{(2-2.5 \times 0.0427)}-\frac{3.59 \times 2.5}{2}$$
P = $$\frac{n \mathrm{RT}}{\mathrm{V}}$$
∴ P = $$\frac{2.5 \times 0.0821 \times 310}{2}$$ = 31.8 atm

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text:
A researcher is studying the decomposition of A as shown by the general reaction below: 2 A(g) 2 B(g) + 2 C(g) Initially, the scientist fills an evacuated 4.956 L flask with 9.570 x 10-1 moles of species A. Upon equilibrium, it is determined that the concentration of A is 1.724 x 10-1 M. Calculate Kc.

Question

A researcher is studying the decomposition of A as shown by the general reaction below:

2 A(g) 2 B(g) + 2 C(g)

Initially, the scientist fills an evacuated 4.956 L flask with 9.570 x 10-1 moles of species A. Upon equilibrium, it is determined that the concentration of A is 1.724 x 10-1 M. Calculate Kc.