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Found in: Page 578

### Fundamentals Of Physics

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

# Does the temperature of an ideal gas increase, decrease, or stay the same during (a) an isothermal expansion, (b) an expansion at constant pressure, (c) an adiabatic expansion, and (d) an increase in pressure at constant volume?

1. The temperature of an ideal gas during an isothermal expansion stays the same.
2. The temperature of an ideal gas during an expansion at constant pressure increases.
3. The temperature of an ideal gas during an adiabatic expansion decreases.
4. The temperature of an ideal gas during an increase in pressure at constant volume increases.

See the step by step solution

## Stating the given data

The gas is an ideal gas.

## Understanding the concept of ideal gas equation for different conditions

Since the temperature remains constant in an isothermal process, the temperature of an ideal gas during an isothermal expansion stays the same. Using the gas law, we can predict the temperature of an ideal gas during an expansion at constant pressure and during an increase in pressure at constant volume. Then, using the first law of thermodynamics, we can predict the temperature of an ideal gas during an adiabatic expansion.

Formulae:

Ideal gas equation, $pV=nRT$ …(i)

Change in internal energy due to first law of thermodynamics,$∆{E}_{int}=Q-W$ …(ii)

Change in internal energy at constant volume, $∆{E}_{int}=n{C}_{v}∆T$ …(iii)

## (a) Calculation of the temperature during isothermal expansion

For an ideal gas at isothermal expansion, the initial temperature and final temperature is the same, so $\Delta T=0.$

Hence, the temperature of an ideal gas during an isothermal expansion stays the same.

## (b) Calculation of the temperature at constant pressure during an expansion

For constant pressure, there is change in volume. Thus, the equation (i) gives the temperature value as follows:

$\begin{array}{c}p\Delta V=nR\Delta T\\ \Delta T=\frac{p\Delta V}{nR}\end{array}$

Since, $\Delta V>0$thus, the value of change in temperature using the above equation is $\Delta T>0$

Hence, the change in temperature increases at constant pressure.

## (c) Calculation of the temperature during an adiabatic expansion

For adiabatic expansion, the total heat is constant, i.e. $Q=0$

Thus, using equation (ii), the change in internal energy is given as follows:

$∆{E}_{int}=-W$

For adiabatic expansion, gas is expanding, so that work done is positive and is increasing.

Thus, $∆{E}_{int}<0$

Now, using this value in equation (iii), we can get the change in temperature as follows:

$\Delta T<0$

Hence, the change in temperature decreases at constant adiabatic expansion.

## (d) Calculation of the temperature during an increase in pressure at constant volume

For constant volume, there is change in pressure. Thus, equation (i) gives the temperature value as follows:

$\begin{array}{c}\Delta pV=nR\Delta T\\ \Delta T=\frac{\Delta pV}{nR}\end{array}$

Since, $\Delta p>0$the change in internal energy is found using the above equation as $\Delta T>0$

Therefore, the change in temperature increases with an increase in the pressure at constant volume.

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