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tokens: ( 1986 ) asand C H is the heat transfer function defined byIn these equations, γ is the specific heat ratio, k B is the Boltzmann constant, μ is the mean molecular weight of a gas particle, m H is the mass of a hydrogen atom, and s a is a ratio of the drift velocity and the thermal gas velocity defined byEquation ( 10 ) can be solved for the dust temperature at which the heating of the dust by the gas is in equilibrium with the radiative cooling,which can be further rewritten as where the C values are constants with respect to s a defined by and α values are functions of s a defined by With this reduction of the equation, Equation ( 18 ) is solved for T dust using the minpack 's hybrd and hybrj routines with the modified Powell method (Moré et al. 1980 ). For the purposes of guiding the algorithm, initial guesses were computed byApplying the temporal evolution of the gas temperature, density, and drift velocity determined in the previous section, we are able to compute the dust temperature across the evolution of the shock. In the pre-shock gas, the gas and dust are assumed to be in thermal equilibrium, or T dust = T gas . In addition, we deviate from the formalism used in Jiménez-Serra et al. ( 2008 ), by setting an additional constraint of the velocity when used to calculate the rate of dust heating. Specifically, we assume that as the gas and grains begin to recouple, the drift velocity will eventually fall below the random thermal velocities of the gas. In the hot gas regime, this thermal velocity may be nontrivial. As such, the velocity of collisions may be more accurately described for these purposes as As can be seen in Figure 2 , the grain temperature is found to initially track with the velocity structure within the shock, peaking at the same corresponding time as the peak velocity owing to adiabatic heating. However, as the gas continues to heat up, the grain temperature undergoes a secondary heating regime owing to the conductive heating from the hot gas. As such, while the sputtering rate (see Section 2.2.3 ) will fall off rather rapidly, the ability for species to redeposit onto the grains is delayed until the gas can cool sufficiently. 2.2.3. Sputtering One major physical process that is common in astrophysical shocks is the nonthermal desorption of molecules from the ice into the gas phase through the collision of moderately fast gas-phase particles with the dust grains (e.g., sputtering). This process is a key mechanism for lifting complex molecules into the gas phase nondestructively. For typical C-shock velocities (e.g., 10–40 km s −1 ), the sputtering is significant enough to desorb most species from the ice mantle without destroying them or having significant grain core erosion, both of which are common in high-velocity J-shocks (Requena-Torres et al. 2006 ). In order to include these processes, the sputtering yields must be computed for each species on the ice, which is dependent on the conditions of the shock, the nature