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Video Abstract

Video Abstract

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BACKGROUND:

A complex set of medical, social, and financial factors underlie decisions to discharge very preterm infants. As care practices change, whether postmenstrual age and weight at discharge have changed is unknown.

METHODS:

Between 2005 and 2018, 824 US Vermont Oxford Network member hospitals reported 314 811 infants 24 to 29 weeks’ gestational age at birth without major congenital abnormalities who survived to discharge from the hospital. Using quantile regression, adjusting for infant characteristics and complexity of hospital course, we estimated differences in median age, weight, and discharge weight z score at discharge stratified by gestational age at birth and by NICU type.

RESULTS:

From 2005 to 2018, postmenstrual age at discharge increased an estimated 8 (compatibility interval [CI]: 8 to 9) days for all infants. For infants initially discharged from the hospital, discharge weight increased an estimated 316 (CI: 308 to 324) grams, and median discharge weight z score increased an estimated 0.19 (CI: 0.18 to 0.20) standard units. Increases occurred within all birth gestational ages and across all NICU types. The proportion of infants discharged home from the hospital on human milk increased, and the proportions of infants discharged home from the hospital on oxygen or a cardiorespiratory monitor decreased.

CONCLUSIONS:

Gestational age and weight at discharge increased steadily from 2005 to 2018 for survivors 24 to 29 weeks’ gestation with undetermined causes, benefits, and costs.

What’s Known on This Subject:

A preterm infant generally reaches physiologic stability and functional maturation at ∼36 weeks’ postmenstrual age. As neonatal intensive care practices have evolved, whether postmenstrual age and weight at discharge have changed is unknown.

Postmenstrual age increased 8 days for all infants, and weight increased 316 g or 0.19 standard units on z score for infants discharged from the hospital across all birth gestational ages and NICU types. Changes in clinical practices may explain these increases.

A complex set of medical, social, and financial factors underlie the decision to discharge a very preterm infant from a NICU.1,2  Of primary concern is an infant’s physiologic stability and functional maturation, which generally occurs at ∼36 weeks’ postmenstrual age.3,4  Equally important are the needs to establish discharge readiness;5  find an infant’s medical home;6  assess housing, psychosocial support, and other social needs;7  and take appropriate steps to follow through to address the social determinants of health.8

Given that, the optimal discharge age and weight for stable preterm infants are largely unknown. Historically, discharge occurred when infants achieved a certain weight.2 However, in 3 randomized trials in which infants who were discharged when physiologically ready were compared with those discharged at a set weight9,10  or by physician discretion,11  no adverse consequences of earlier discharge were found. The savings associated with earlier discharge were substantial, from $18 560 per infant in 198610 or$43 686 in 2020 dollars,12  to $5016 per infant in 200213 or$7307 in 2020 dollars,12  and the effect of prolonged hospital stays on families is equally as important as financial interests. However, these trials were conducted in an earlier era of neonatal intensive care, when treatment practices and survival rates for extremely preterm infants were much different from how they are today.

We have observed reductions in neonatal mortality and morbidities,14,15  increases in the use of evidence-based care practices,16  and increases in growth velocity during the birth hospitalization17  since 2000. We sought to describe changes in discharge weight and age for very preterm infants in the United States from 2005 to 2018.

Vermont Oxford Network (VON) is a voluntary worldwide community of practice dedicated to improving the quality, safety and value of newborn care through a coordinated program of data-driven quality improvement, education, and research. The University of Vermont institutional review board determined that use of data from the VON Research Repository for this analysis was not human subjects research.

US hospitals with NICUs contributed data on infants 401 to 1500 g or 22 0/7 to 29 6/7 weeks’ gestational age at birth who were inborn or transferred to the reporting hospital within 28 days of life from January 1, 2005, to December 31, 2018. All data were collected by local staff using standard definitions.18

We included all live-born infants born at 24 0/7 to 29 6/7 weeks’ gestation without major congenital anomalies who survived to discharge from the hospital. We calculated postmenstrual age at discharge as gestational age at birth in days plus total length of stay in days. Transferred infants’ length of stay was tracked by the reporting hospitals until ultimate disposition. We assessed discharge weight only for infants initially discharged from the hospital because we did not have weight at ultimate disposition for transferred infants. The z scores for weight at birth and discharge up to 50 weeks’ postmenstrual age were calculated by using the Fenton growth chart.19  The z scores for discharge after 50 weeks’ postmenstrual age were calculated by using the World Health Organization Child Growth Standards with corrected age.20

To evaluate clinical factors that may influence discharge age, we evaluated, separately, the proportions of infants who were discharged from the hospital on oxygen, on cardiorespiratory monitors, or on any human milk. We also evaluated the proportion of infants transferred to assess whether changes in transfer could explain observed changes.

Hospital characteristics were derived from the VON annual member survey. By using responses to whether the center was required by state regulation to transfer infants to another center for assisted ventilation on the basis of the infant characteristics or duration of ventilation required or whether 1 of 8 surgeries was performed at the center (omphalocele repair, ventriculoperitoneal shunt, tracheoesophageal fistula and/or esophageal atresia repair, bowel resection and/or reanastomosis, meningomyelocele repair, PDA ligation, cardiac catheterization, or cardiac surgery requiring bypass), centers were divided into 4 groups: ventilation restrictions; no surgery; surgery except cardiac requiring bypass; and all surgery. Teaching hospitals included those in which pediatric residents, neonatology fellows, or other residents (family practice, obstetrics, anesthesia, etc) participated in direct patient care in the NICU.

We stratified infants by week of gestational age at birth and evaluated the median age and weight at discharge from 2005 to 2018 (Figs 1 and 2). We used quantile regression21,22  to estimate the annual change in median age, weight, and weight z score at discharge, adjusting for differences in infant and delivery characteristics, including year of birth, gestational age at birth, 1-minute Apgar score, small size for gestational age, cesarean delivery, sex, multiple gestation, and birth location (inborn or outborn). We estimated changes in medians over the 14 years with 95% compatibility intervals (CIs).23  We stratified models by week of gestational age and by NICU type to estimate differences over time within these categories.

FIGURE 1

Median gestational age at discharge in weeks for infants born at 24 to 29 weeks’ gestational age who survived to discharge from the hospital, 2005–2018.

FIGURE 1

Median gestational age at discharge in weeks for infants born at 24 to 29 weeks’ gestational age who survived to discharge from the hospital, 2005–2018.

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FIGURE 2

Median weight at discharge in grams for infants born at 24 to 29 weeks’ gestational age who were initially discharged from the hospital, 2005–2018.

FIGURE 2

Median weight at discharge in grams for infants born at 24 to 29 weeks’ gestational age who were initially discharged from the hospital, 2005–2018.

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TABLE 1

Comparison of Infant Characteristics, 2005 and 2018

2005 Infants, n = 19 4692018 Infants, n = 21 998
At birth
Gestational age, mean ± SDa 27.1 ± 1.6 27.1 ± 1.6
1-min Apgar score, median (IQR)a 6 (4–7) 5 (3–7)
Small for gestational age, n (%)a 1182 (6.1) 1674 (7.6)
Cesarean delivery, n (%)a 13 426 (69.0) 15 788 (71.8)
Male sex, n (%)a 10 155 (52.2) 11 350 (51.6)
Multiple gestation, n (%)a 5121 (26.3) 5015 (22.8)
Inborn at reporting hospital, n (%)a 16 032 (82.3) 19 064 (86.7)
Neonatal course, n (%)
Delivery room interventiona,b 16 091 (82.7) 17 203 (78.2)
Ventilation after initial resuscitationa 16 207 (83.3) 14 926 (67.9)
Any surgerya 4561 (23.4) 3813 (17.3)
Morbidities, n (%)
Respiratory distress syndromea 17 035 (87.5) 18 452 (83.9)
Early bacterial sepsis 415 (2.1) 247 (1.1)
Any late infection 5100 (26.3) 1968 (9.0)
Severe intraventricular hemorrhage 1545 (8.1) 1431 (6.7)
Necrotizing enterocolitis or gastrointestinal perforation 1663 (8.6) 1337 (6.1)
Pneumothorax 792 (4.1) 890 (4.0)
Chronic lung disease 7171 (38.3) 7299 (34.0)
2005 Infants, n = 19 4692018 Infants, n = 21 998
At birth
Gestational age, mean ± SDa 27.1 ± 1.6 27.1 ± 1.6
1-min Apgar score, median (IQR)a 6 (4–7) 5 (3–7)
Small for gestational age, n (%)a 1182 (6.1) 1674 (7.6)
Cesarean delivery, n (%)a 13 426 (69.0) 15 788 (71.8)
Male sex, n (%)a 10 155 (52.2) 11 350 (51.6)
Multiple gestation, n (%)a 5121 (26.3) 5015 (22.8)
Inborn at reporting hospital, n (%)a 16 032 (82.3) 19 064 (86.7)
Neonatal course, n (%)
Delivery room interventiona,b 16 091 (82.7) 17 203 (78.2)
Ventilation after initial resuscitationa 16 207 (83.3) 14 926 (67.9)
Any surgerya 4561 (23.4) 3813 (17.3)
Morbidities, n (%)
Respiratory distress syndromea 17 035 (87.5) 18 452 (83.9)
Early bacterial sepsis 415 (2.1) 247 (1.1)
Any late infection 5100 (26.3) 1968 (9.0)
Severe intraventricular hemorrhage 1545 (8.1) 1431 (6.7)
Necrotizing enterocolitis or gastrointestinal perforation 1663 (8.6) 1337 (6.1)
Pneumothorax 792 (4.1) 890 (4.0)
Chronic lung disease 7171 (38.3) 7299 (34.0)
a

b

Includes face mask ventilation, endotracheal tube ventilation, cardiac compression, epinephrine, or surfactant in the delivery room.

Over the 14 years, 824 hospitals contributed data, of which 47% were teaching hospitals, 72% were nonprofit, 51% had surgical capabilities, and 8% were freestanding children’s hospitals. Overall, 314 811 infants met eligibility criteria. During the study period, hospitals with restrictions on ventilation contributed 1.5% of infants, hospitals that did not do surgery contributed 16%, hospitals that did surgery except cardiac surgery requiring bypass contributed 52%, and hospitals that did all surgeries contributed 31%. Characteristics of 2005 and 2018 cohorts that are included in the quantile regression models are in Table 1.

TABLE 2

Adjusted Differences in Median Discharge Age in Days and Weight in Grams, With 95% CIs, From 2005 to 2018

Change in Discharge Age (CI), dChange in Discharge Wt (CI), gChange in Discharge Wt (CI), z Score
Overall 8 (8 to 9) 316 (308 to 324) 0.19 (0.18 to 0.20)
Gestational age at birth, wk
24 13 (12 to 14) 533 (492 to 574) 0.49 (0.44 to 0.54)
25 10 (9 to 11) 434 (402 to 465) 0.42 (0.38 to 0.46)
26 9 (8 to 10) 376 (351 to 400) 0.29 (0.25 to 0.32)
27 8 (8 to 9) 315 (296 to 334) 0.20 (0.17 to 0.22)
28 8 (7 to 8) 288 (273 to 304) 0.13 (0.11 to 0.15)
29 7 (6 to 7) 248 (235 to 260) 0.08 (0.07 to 0.10)
NICU type
Restrictions on ventilation 8 (7 to 10) 279 (224 to 335) 0.05 (−0.03 to 0.12)
No surgery 8 (8 to 9) 278 (260 to 296) 0.09 (0.07 to 0.11)
No restrictions on ventilation and no surgery except cardiac surgery requiring bypass 8 (8 to 8) 321 (310 to 331) 0.21 (0.20 to 0.22)
No restrictions on ventilation and all surgery, including cardiac surgery requiring bypass 10 (9 to 10) 365 (349 to 381) 0.19 (0.17 to 0.21)
Change in Discharge Age (CI), dChange in Discharge Wt (CI), gChange in Discharge Wt (CI), z Score
Overall 8 (8 to 9) 316 (308 to 324) 0.19 (0.18 to 0.20)
Gestational age at birth, wk
24 13 (12 to 14) 533 (492 to 574) 0.49 (0.44 to 0.54)
25 10 (9 to 11) 434 (402 to 465) 0.42 (0.38 to 0.46)
26 9 (8 to 10) 376 (351 to 400) 0.29 (0.25 to 0.32)
27 8 (8 to 9) 315 (296 to 334) 0.20 (0.17 to 0.22)
28 8 (7 to 8) 288 (273 to 304) 0.13 (0.11 to 0.15)
29 7 (6 to 7) 248 (235 to 260) 0.08 (0.07 to 0.10)
NICU type
Restrictions on ventilation 8 (7 to 10) 279 (224 to 335) 0.05 (−0.03 to 0.12)
No surgery 8 (8 to 9) 278 (260 to 296) 0.09 (0.07 to 0.11)
No restrictions on ventilation and no surgery except cardiac surgery requiring bypass 8 (8 to 8) 321 (310 to 331) 0.21 (0.20 to 0.22)
No restrictions on ventilation and all surgery, including cardiac surgery requiring bypass 10 (9 to 10) 365 (349 to 381) 0.19 (0.17 to 0.21)

The median postmenstrual age at discharge from the hospital for the overall population was 38.3 weeks (interquartile range [IQR]: 36.6–40.7). Figure 1 reveals the median postmenstrual age at discharge by gestational age at birth and birth year from 2005 to 2018. Over the 14 years, median unadjusted postmenstrual age at discharge increased 9 days, and median adjusted age increased an estimated 8 days (95% CI: 8 to 9). The risk-adjusted estimates of the differences from quantile regression models by gestational age and NICU type are in Table 2.

The median weight at discharge from the hospital for the 273 109 infants initially discharged from the hospital was 2600 g (IQR: 2260–3078). Figure 2 reveals the median weight at discharge by gestational age at birth and birth year from 2005 to 2018. Over the 14 years, median unadjusted weight at discharge increased 360 g and median adjusted weight at discharge increased an estimated 316 g (95% CI: 308 to 324). Risk-adjusted estimates by gestational age and NICU type are in Table 2.

Figure 3 reveals the median z score for weight at discharge by gestational age at birth and birth year from 2005 to 2018. Over the 14 years, median unadjusted z score for weight increased 0.22 standard units and median adjusted z score for weight at discharge increased an estimated 0.19 standard units (95% CI: 0.18 to 0.20). Risk-adjusted estimates by gestational age and NICU type are in Table 2.

FIGURE 3

Median z score for weight at discharge for infants born at 24 to 29 weeks’ gestational age who were initially discharged from the hospital, 2005–2018.

FIGURE 3

Median z score for weight at discharge for infants born at 24 to 29 weeks’ gestational age who were initially discharged from the hospital, 2005–2018.

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From 2005 to 2018, the proportion of infants who were discharged from the hospital on any human milk increased from 40% to 48%. Among infants initially discharged from the hospital, the proportion discharged on a cardiorespiratory monitor decreased from 49% to 26%, whereas those discharged on oxygen decreased from 27% to 22%. The proportion of infants who were never transferred increased from 71% to 76%.

In this large cohort of infants born 24 to 29 weeks’ gestational age and admitted to 824 NICUs in the United States who survived to initial discharge, adjusted postmenstrual discharge age increased by 8 days (95% CI: 8 to 9) from 2005 to 2018, adjusted discharge weight increased by 316 g (95% CI: 308 to 324), and adjusted median z score for weight at discharge increased by 0.19 standard units (95% CI: 0.18 to 0.20) across all gestational ages and NICU types. If 1 day in the NICU costs $3000,24 or$3572 in 2020 dollars,12  then an extra 8 days could cost an additional \$28 576 if every day of a NICU stay costs the same amount. Moreover, this increase means that families and infants are separated for >1 week longer today than they were 14 years ago. Whether the increased separation leads to better preparation for discharge is unknown because optimal discharge age and weight are largely unknown.

Before adjustment, median postmenstrual age at discharge increased 9 days and median discharge weight increased 360 g. More infants survived in 2018 than in 2005, and it is possible that infants who would not have survived in 2005 were surviving in 2018 and required more time to reach physiologic maturity before discharge. However, after adjusting for infant characteristics at birth, the gestational age and weight at discharge still increased dramatically over the time period.

The proportion of infants discharged from the hospital on any human milk, which includes exclusive human milk and human milk plus fortifier, increased from 2005 to 2018. Nutritional management quality improvement efforts2529  and the Baby-Friendly Hospital Initiative30  may have contributed to the increased discharge weight and decreased growth failure, although we do not know whether similar improvements in nutrition and human milk could be achieved in shorter stays or whether the improvements were a result of the longer stays. The American Academy of Pediatrics recommends human milk31  because it protects against life-threatening complications of preterm birth.32  We previously found that infants discharged on human milk alone or in combination with formula showed improvements in weight z score change and weight gain velocity from 2012 to 2016 similar to formula-fed infants.33  Still, disparities by race and ethnicity persist, with non-Hispanic Black and Native American infants less likely than non-Hispanic white infants to be discharged on any human milk.33

Fewer infants were discharged from the hospital with respiratory monitoring and oxygen support. Apnea of prematurity and bradycardia events, which typically resolve between 37 and 43 weeks’ postmenstrual age,3,34,35  require observation around the time of discharge. However, in a study of >1400 premature infants, Lorch et al4  found that many did not have future apnea or bradycardia events when they were otherwise ready for discharge. They suggested that gestational age at birth, chronological age at observation, and days off methylxanthines should be considered when deciding whether an infant will be event-free at discharge.4  Instead, they observed that most NICUs have protocols determining the length of observation needed for apnea and bradycardia at the time of discharge. One study found this observation time ranged from 1 to 21 days,36  although the majority of neonatologists require a 5 to 7-day observation period without apnea or bradycardia events before discharge.37  Whether infants were discharged later to establish cardiorespiratory maturity or because of policies governing the number of days infants must be observed to be apnea free before discharge is unknown. The work of Walsh et al38,39  on timed room-air challenges for bronchopulmonary dysplasia were published just before our study period began and may play a role in increased length of stay, better growth, and decreased rates of discharge on oxygen support.40

This study’s limitations include lack of information on discharge policies at the NICU level, which would give us additional insights into why we observed these increases. Infants may be staying longer because of misaligned financial incentives; when payment is based on fee for service, physician reimbursement and other incentives may lead to longer stays. We do not know to what extent financial incentives may be playing a role. We also do not know staffing issues at the NICU level that might influence timing of discharge. For example, Profit et al41  found that NICU census and the patient-nurse ratio influenced discharge of moderately preterm infants, whereas Silber et al13  observed a decreased rate of discharge on weekends and increased rates on Mondays and Fridays. We do not know readmission rates for infants, and we do not have enough information to determine causal mechanisms for increased discharge age or weight. There could be unmeasured differences in survivors that could contribute to the findings.

Gestational age and weight at discharge increased steadily from 2005 to 2018 for infants 24 to 29 weeks’ gestation. The benefits, costs, and harms of increased age and weight at discharge are undetermined. Nonetheless, families and infants are separated for more than a week longer today than they were 14 years ago. The effects of prolonged separation cannot be underestimated.

We are indebted to our colleagues who submit data to VON on behalf of infants and their families. The list of centers contributing data to this study are in Supplemental Table 3.

Dr Edwards drafted the initial manuscript; Ms Greenberg conducted the data analyses; and all authors conceptualized and designed the study, reviewed and revised the manuscript, and approved the final manuscript as submitted.

FUNDING: Funded by Vermont Oxford Network.

• CI

compatibility interval

•
• IQR

interquartile range

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## Competing Interests

POTENTIAL CONFLICT OF INTEREST: The authors have indicated they have no potential conflicts of interest to disclose.

FINANCIAL DISCLOSURES: Drs Edwards and Ehret receive salary support through a grant from Vermont Oxford Network to The University of Vermont. Ms Greenberg is an employee of Vermont Oxford Network. Dr Horbar is chief executive officer, president, chief scientific officer, and an unpaid member of the Board of Directors of Vermont Oxford Network.


text:
# Choose the correct answer. A thermodynamic state function is a quantity

+1 vote
152 views

edited

Choose the correct answer. A thermodynamic state function is a quantity

(i) used to determine heat changes

(ii)whose value is independent of path

(iii) used to determine pressure volume work

(iv) whose value depends on temperature only

+1 vote
by (128k points)
selected by

A thermodynamic state function is a quantity whose value is independent of a path.

Functions like p, V, T etc. depend only on the state of a system and not on the path.

Hence, alternative (ii) is correct.

format: plain


text:
# Proof of extended supermartingale convergence theorem

There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic Programming" the theorem is:

"Let $$Y_t, X_t, Z_t, t = 1,2,3,....$$ be three sequences of random variables and let $$\mathcal{F_t}$$ be sets of random variables such that $$\mathcal{F_t} \subset \mathcal{F_{t+1}}$$ for all t, suppose that:

(a) The random variables $$Y_t, X_t, Z_t$$ are nonnegative and are functions of the random variables in $$\mathcal{F}_t$$

(b) For each $$t$$ we have $$E[Y_{t+1}|\mathcal{F_t}] \leq Y_t - X_t +Z_t$$

(c) $$\sum_{t=0}^\infty Z_t \lt \infty$$

Then:

$$\sum_{t=0}^{\infty}X_t \lt \infty$$ and there exists a nonnegative random variable $$Y$$ such that $$Y_t \rightarrow Y$$ with probability 1."

The problem is, I can't find any proofs of this anywhere. Most texts on probability theory prove the standard Martingale Convergence Theorem but I feel this is a big step from that.

Can anyone direct me to a source which proves the above or write out a proof from the standard convergence theorem?

Here's one approach.

First notice that $$R_t: = Y_{t} + \sum_{i=1}^{t-1} X_i - \sum_{i=1}^{t-1} Z_i$$ is a supermartingale, since $$R_{t+1} - R_t = Y_{t+1} - Y_t + X_t - Z_t,$$ giving $$E(R_{t+1} - R_t | \mathcal{F}_t) = E (Y_{t+1} | \mathcal{F}_t) - Y_t + X_t - Z_t$$ which is less than or equal to $$0$$ with probability $$1$$, by (b).

Now, we don't have a fixed lower bound for the supermartingale $$R$$, so we can't apply the convergence theorem directly.

However, for any $$a>0$$, consider the stopping time $$\tau_a = \inf\{t: \sum_{i=1}^t Z_i>a\},$$ with $$\tau_a=\infty$$ if $$\sum_{i=1}^t Z_i\leq a$$ for all $$t$$.

We can define $$R^{(a)}(t):=R(t\wedge \tau_a)= \begin{cases} R_t&\text{ if }t<\tau_a\\ R_{\tau_a}&\text{ if }t\geq \tau_a \end{cases}.$$

$$R^{(a)}$$ is also a supermartingale for any $$a$$, and $$R^{(a)}(t)$$ is bounded below by $$-a$$.

So by the martingale convergence theorem, for any given $$a$$, $$R^{(a)}(t)$$ converges to some finite limit with probability $$1$$. By countable additivity, we get that with probability $$1$$, $$R^{(a)}(t)$$ converges to a finite limit for all $$a\in\mathbb{Z}$$.

But if $$\sum_{i=0}^\infty Z_i<\infty$$, which from (c) we assume happens with probability $$1$$, then for all large enough $$a\in\mathbb{Z}$$, we have $$\tau_a=\infty$$, and so $$R^{(a)}(t)=R(t)$$ for all $$t$$. Since we know $$R^{(a)}(t)$$ converges, we also get that $$R(t)$$ converges.

Finally, since $$R(t)$$ converges and $$\sum_{i=0}^{t-1} Z_i$$ converges, we also have that $$Y_t+\sum_{i=1}^{t-1} X_i$$ converges. Since $$\sum_{i=1}^{t-1} X_i$$ is non-decreasing in $$t$$, and $$Y_t$$ is non-negative for all $$t$$, the only way this can happen is if $$Y_t$$ and $$\sum_{i=1}^{t-1} X_i$$ both converge, as required.

• Thanks, excellent! After a little digging one other alternative way is to reference a super-martingale convergence theorem for the sum missing the $-X_t$ term to get convergence and then this forces the $\sum X_t$ to be finite A.S. – FourierFlux Apr 19 '20 at 7:10


text:
# How do you factor #x^2 + x + 1 ?

May 4, 2015

To factor ${x}^{2} + x + 1$
use the quadratic formula for roots
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

In this case:
$x = \frac{- 1 \pm \sqrt{{1}^{2} - 4 \left(1\right) \left(1\right)}}{2 \left(1\right)}$

$x = - \frac{1}{2} \pm \frac{\sqrt{- 3}}{2}$

With no Real roots
and thus no factors if we are restricted to Real numbers.

However, if we are allowed to venture into Complex numbers
$x = - \frac{1}{2} \pm \frac{\sqrt{3}}{2} i$

and the factors are
$\left(x + \frac{1}{2} + \frac{\sqrt{3}}{2} i\right) \left(x + \frac{1}{2} - \frac{\sqrt{3}}{2}\right)$


text:
# Calculate the ratio of the concentration of acetate and acetic acid

Calculate the ratio of the concentration of acetate and acetic acid

## What is the ratio of the concentration of acetate and acetic acid required in a buffer system of pH 5.3?

Buffer acetate comprising a mixture of CH3COOH and CH3COONa. pKa of CH3COOH = 4.74.

CH3COONa ⇌Na+ + CH3COO-

CH3COOH ⇌ H+ + CH3COO-

Calculation of pH acetate buffer,

Let, [CH3COO-] and [CH3COOH] denotes [A-] and [HA], respectively.

pH = pKa + log [A-]/[HA]

5.3 = 4.74 + log [A-]/[HA]

log [A-]/[HA] = 5.3 – 4.74

log [A-]/[HA] = 0.56

[A-]/[HA] = antilog 0.56

[A-]/[HA] = 3.63

So, the ratio of the concentration of acetate and acetic acid is 3.63 : 1.

## Calculate the ratio of acetate to acetic acid for an acetic acid/acetate buffer to which strong base was added to result in a pH of 5.76? Ka=1.8×10−5.

Well, the weak acid and conjugate weak base form a buffer…

HOAc  weak acid
OAc  conjugate weak base

And for buffers, the Henderson-Hasselbalch equation applies.

pH=pKa+log[A][HA]

The ratio of acetate to acetic acid is given by [OAc][HOAc].

pHpKa=log[OAc][HOAc]

Thus, with the Ka of acetic acid as 1.8×105, the pKa is log(Ka)=4.76, and:

10pHpKa=[OAc][HOAc]

=105.764.76

=10.0

So there is about a 10-fold excess of acetate compared to acetic acid. That should make sense, since the pH is higher than the pKa, i.e. the solution is basic so that the conjugate weak base species dominates.

From this we can say that for every ΔpH=±1, the concentration of one of the species in the buffer multiplies by 10.

## What Is The Ratio Of Acetate Ion To Acetic Acid?

So the ratio of the concentration of acetate and acetic acid is 3.63 : 1. So log (AcO-/AcOH) = pH- pKa= 5.3000- 4.7447 =0.5553 . So (Acetate /Acetic Acid) =10^(0.5553) =3.592.

## What Is The Ratio Of Acetate Ions To Acetic Acid Called?

And for buffers the Henderson-Hasselbalch equation applies. The ratio of acetate to acetic acid is given by [OAc−][HOAc] . So there is about a 10 -fold excess of acetate compared to acetic acid.

## What Is The Ratio Of Acetic Acid?

Acetic acid: Pure laboratory-grade acetic acid. To satisfy the expression the ratio of [C2H3O2]/[HC2H3O2] must be 0.36 to 1.

## What is the ratio of the concentration of acetic acid to sodium acetate in the solution?

In a buffer solution with a pH of 5.76, the ratio of acetic acid to sodium acetate in terms of molar concentration is 10 to 1.

## What kind of connection do acetic acid and sodium acetate have with one another?

There is an acetate ion present in the acetic acid. Another kind of acetate ion may be found in sodium acetate. Because acetic acid and sodium acetate share a common ion, the acetate ion, the addition of sodium acetate may change the equilibrium of acetic acid. There is an acetate ion present in the acetic acid.

## What Is Acetate Formula?

Acetate | C2H3O2– – PubChem.

## What does “Buffer Ratio” stand for?

The number of moles of the weak acid and its conjugate base must be relatively high in comparison to the number of moles of strong acid or base that may be added for the buffer to be useful…. When the ratio falls somewhere between 10:1 and 1:10, buffers are regarded to be operating at an effective level.

## Where Can I Find Information Regarding The Concentrations Of Acetate And Acetic Acid In A Buffer Solution?

In order to make a buffer solution, equal amounts of 0.200 M acetic acid and 0.600 M sodium acetate are combined in a container and shaken. For the value of Ka for acetic acid, use 1.80 x 10-5.

## When the PKa PH is taken into consideration, what is the ratio of acetic acid to its conjugate base?

For instance, given that the pKa value of acetic acid is around 4.7, when the pH is the same as the pKa value, the ratio of acetate to acetic acid would be one to one. When the pH is at 4.0, the ratio of fluoride ion to hydrofluoric acid would be 1:1 for an additional acid such as HF, which has a pKa value of around 4.0. When the pH is at 4.0, this ratio holds true.

## How can I figure out the value of Ka?

As was said before, [H3O+] equals a pH of 10. If you already know the pH of the solution and you know that x = [H3O+], then you may write x = 10–2.4. The search for a numerical value for Ka is now feasible. Ka = (10–2.4)2 /(0.9 – 10–2.4) = 1.8 x 10–5.

## What exactly is acetate buffer used for?

As addition as being used in the purification and precipitation of nucleic acids, sodium acetate buffers are also utilized in the crystallization of proteins and the coloring of gels that are utilized in protein electrophoresis. Acetate buffers are not only easy to make and cheap, but they may also be kept at room temperature once they have been prepared.

## What Is the Ph Value of Sodium Acetate and Acetic Acid?

The pH of a sodium acetate-acetic acid buffer is 4.50, and this is the problem. Determine the ratio of [CH3COO-] to [CH3COOH] using a calculator.

## What Reaction Occurs When Acetate Is Added To A Solution That Already Contains Acetic Acid?

When acetate is introduced into an acetic acid system that is already at equilibrium, it will react with some of the hydronium ions present, which will cause the equilibrium to move to the left. Since the concentration of hydronium ions is decreasing, one would expect the pH to rise (become less acidic).

## What Happens To The Degree Of Ionization Of Acetic Acid When Sodium Acetate Is Added To Acetic Acid?

The presence of acetate ion causes an increase in the degree to which acetic acid is ionized when sodium acetate is added to it. This is because acetic acid already contains acetate ion.

## What Reaction Occurs When Sodium Acetate Is Added to a Solution Containing Acetic Acid?

When sodium acetate is added to a solution of acetic acid, the concentration of $H+$ ions in the buffer solution falls. At the same time, the concentration of salt in the buffer solution rises, which causes the pH value to rise as a result.

## Is It True That Acetate Is the Same Thing as Acetic Acid?

The primary distinction between acetic acid and acetate lies in the fact that acetic acid is an element with no net electrical charge, while acetate is an anion with a net negative charge. Acetate ion is the conjugate base of acetic acid, which is an organic chemical that is used in the production of vinegar. Acetic acid is one of the components that contributes to the production of vinegar.

## F.A.Q calculate the ratio of the concentration of acetate and acetic acid

### What is the ratio of acetate ions to acetic acid?

The ratio of the acetate ion to acetic acid is 0.550: 1.

### What are the concentrations of acetic acid and acetate in a buffer solution?

A buffer solution is prepared from equal volumes of 0.200 M acetic acid and 0.600 M sodium acetate. Use 1.80 × 10−5 as Ka for acetic acid.

### What is the ratio of concentration of acetic acid and acetate ions for a buffer solution with pH of 5.74 pKa of acetic acid is 4.74 )*?

The ratio of concentration of acetic acid and acetate ion to have in solution of pH 5.74 is 1: 10.

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text:
# What are resistor combinations?

##### 1 Answer
May 28, 2014

Resistor combinations combine series and parallel paths together in a single circuit.

This is a fairly simple combination circuit. To solve any combination circuit, simplify it down to a single series circuit. This is usually done most easily starting at the farthest point from the power source.

On this circuit, find the equivalent resistance of ${R}_{2}$ and ${R}_{3}$, as if they were a single resistor connected to the others in series.

$\frac{1}{R} _ T = \frac{1}{R} _ 2 + \frac{1}{R} _ 3$
$\frac{1}{R} _ T = \frac{1}{30} + \frac{1}{50}$
$\frac{1}{R} _ T = \frac{8}{150}$
Take the reciprocal of each to get ${R}_{T}$ out of the denominator:
${R}_{T} = \frac{150}{8}$
${R}_{T} = 18.75 \Omega$

Now add this to the $20 \Omega$ of ${R}_{1}$ and the $20 \Omega$ of ${R}_{2}$ to get a total resistance on this circuit of $58.75 \Omega$.

Other circuits may be more complex, and you may have to do this several times - going back and forth between series and parallel - until this is simplified down to a simple series or parallel circuit.


text:
mersenneforum.org Getting a new Prime to calculate
 Register FAQ Search Today's Posts Mark Forums Read

 2008-12-16, 16:16 #1 roemer2201   Dec 2008 127.0.0.1 E16 Posts Getting a new Prime to calculate Hello :-) I got a PC with a AMD Athlon 1200+ and he recived a relativly huge prime to calculate. One iteration to about 0.405 seconds, if the PC runs 24/7 it would take 219 days... I think it would be better if my CPU calculates a smaller number. So I want to know how i can stop calculating the current number and recive a new smaller one without deleting the worktodo.ini. And maybe i can recive the GHZ-days for the calculating so far. How can i do that? Last fiddled with by roemer2201 on 2008-12-16 at 16:16
2008-12-16, 18:15   #2
petrw1
1976 Toyota Corona years forever!

"Wayne"
Nov 2006

32×5×103 Posts

Quote:
 I got a PC with a AMD Athlon 1200+ and he recived a relativly huge prime to calculate. One iteration to about 0.405 seconds, if the PC runs 24/7 it would take 219 days...
If you do not specify otherwise you will get LL tests on a PC like yours that will probably take that long.

Quote:
 I think it would be better if my CPU calculates a smaller number.
You can specifically request Double Checks ... they will probably take about 1/4 as long. Or any of the several types of factoring assignments which would finish in hours or days.
Code:
Test...
Worker Windows...
Type of Work to get
Quote:
 So I want to know how i can stop calculating the current number and recive a new smaller one without deleting the worktodo.ini. And maybe i can recive the GHZ-days for the calculating so far. How can i do that?
Code:
Advanced...
Unreserve Exponent...
Partial credit is NOT awarded. The assignment must complete.

2008-12-20, 16:12   #3
Mr. P-1

Jun 2003

100100100012 Posts

Quote:
 Originally Posted by roemer2201 Hello :-) I got a PC with a AMD Athlon 1200+ and he recived a relativly huge prime to calculate. One iteration to about 0.405 seconds, if the PC runs 24/7 it would take 219 days... I think it would be better if my CPU calculates a smaller number. So I want to know how i can stop calculating the current number and recive a new smaller one without deleting the worktodo.ini. And maybe i can recive the GHZ-days for the calculating so far. How can i do that?
What work are you actually doing on it currently? First-time LL tests usually require P-1 factoring, and may also require some trial factoring prior to starting the LL test proper. This preliminary work takes a few hours or days and you will get credit if you finish it.

If you have started the LL test proper, and you do not complete it then the only way to save the work you have done so far would be to find someone else willing to take it on, and send them the save files. They, not you, would then get the credit for the work you did.

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