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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# A sample of an ideal gas is taken through the cyclic process abca as shown in figure. The scale of the vertical axis is set by ${\text{p}}_{\text{b}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{5}\text{\hspace{0.17em}}\mathbf{kPa}$ and ${{\text{p}}}_{{\text{ac}}}{\mathbf{=}}{\mathbf{2}}{\mathbf{.}}{\mathbf{5}}{\text{\hspace{0.17em}}}{\mathbf{kPa}}$. At point a,${\text{T}}{\mathbf{=}}{\mathbf{200}}{\text{\hspace{0.17em}}}{\mathbf{K}}$ .aHow many moles of gas are in the sample?What are:b.The temperature of gas at point bc.Temperature of gas at point cd.The net energy added to the gas as heat during the cycle?

1. The number of moles of gas present in the sample are $\text{1.5 mol}$.
2. The temperature of the gas at point b is .$1.8×{10}^{3}\text{K}$
3. The temperature of the gas at point c is $6.0×{10}^{2}\text{K}$.
4. The net energy added to the gas as heat during the cycle is $5.0×{10}^{3}\text{J}.$

See the step by step solution

## Step 1: Given

${P}_{b}=7.5\text{kPa}=7500\text{Pa}\phantom{\rule{0ex}{0ex}}{P}_{ac}=2.5\text{\hspace{0.17em}kPa}=2500\text{\hspace{0.17em}Pa}$

1. At point a, $T=200\text{\hspace{0.17em}K}$
2. At point a, $V=1{\text{\hspace{0.17em}m}}^{3}$
3. At point b, $V=3{\text{\hspace{0.17em}m}}^{3}$

## Step 2: Determine the concept

Find the number of moles and the temperature of the gas at points b and c present in the sample using the gas law. Net energy added can be found from the net work done by the gas using the given graph.

Formula is as follow:

${p}_{i}{v}_{i}=nR{T}_{i}$

Here, p is pressure, v is volume, T is temperature, R is universal gas constant and n is number of moles.

## Step 3: (a) Determine the number of moles of gas present in the sample

According to the gas law,

$PV=nRT\phantom{\rule{0ex}{0ex}}n=\frac{PV}{RT}$

From the given graph, at point a,

$n=\frac{2500\left(1\right)}{8.314\left(200\right)}\phantom{\rule{0ex}{0ex}}n=1.5\text{\hspace{0.17em}mol}\phantom{\rule{0ex}{0ex}}$

Hence, the number of moles of gas present in the sample are .$\text{1.5 mol}$

## Step 4: (b) Determine the temperature of the gas at point b

By gas law,

$\frac{{P}_{b}{V}_{b}}{{P}_{a}{V}_{a}}=\frac{{T}_{b}}{{T}_{a}}$

${T}_{b}=\frac{{P}_{b}{V}_{b}}{{P}_{a}{V}_{a}}{T}_{a}\phantom{\rule{0ex}{0ex}}{T}_{b}=\frac{7500\left(3\right)}{2500\left(1\right)}\left(200\right)$

${T}_{b}=1.8×{10}^{3}\text{K}$

Hence, the temperature of the gas at point b is $1.8×{10}^{3}\text{K}$.

## Step 5: (c) Determine the temperature of the gas at point c

Similarly,

${T}_{c}=\frac{{P}_{c}{V}_{c}}{{P}_{a}{V}_{a}}{T}_{a}\phantom{\rule{0ex}{0ex}}{T}_{c}=\frac{2500\left(3\right)}{2500\left(1\right)}\left(200\right)\phantom{\rule{0ex}{0ex}}$

${T}_{c}=600=6.0×{10}^{2}\text{K}$

Hence, the temperature of the gas at point c is $6.0×{10}^{2}\text{K}$.

## Step 6: (d) Determine the net energy added to the gas as heat during the cycle

First law of thermodynamics gives,

$△E=Q-W$

For isothermal process,

$Q=W\phantom{\rule{0ex}{0ex}}Q=\text{Area under the curve}\phantom{\rule{0ex}{0ex}}Q=\frac{1}{2}\left(\text{ac}\right)\left(\text{bc}\right)$

Substitute the values as:

$Q=\frac{1}{2}\left(2\right)\left(7500-2500\right)\phantom{\rule{0ex}{0ex}}$

$Q=5000=5.0×{10}^{3}\text{J}$

Hence, the net energy added to the gas as heat during the cycle is $5.0×{10}^{3}\text{J}.$

Therefore, the number of moles of gas present in the sample,temperature at a point, and heat added can be foundusing gas law.