A $3.65-\mathrm{mol}$ sample of an ideal diatomic gas expands adiabatically from a volume of $0.1210 \mathrm{~m}^3$ to $0.750 \mathrm{~m}^3$. Initially the pressure was $1.00 \mathrm{~atm}$. Determine the work done on the gas. (Assume no molecular vibration.)

A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of $5.00 \mathrm{~atm}$ and a volume of $12.0 \mathrm{~L}$ to a final volume of $30.0 \mathrm{~L}$. Find $Q, W$, and $\Delta E_{\text {int }}$.

A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of $5.00 \mathrm{~atm}$ and a volume of $12.0 \mathrm{~L}$ to a final volume of $30.0 \mathrm{~L}$. What are the initial and final temperatures?

A 1.00-mol sample of an ideal diatomic gas, originally at $1.00 \mathrm{~atm}$ and $20^{\circ} \mathrm{C}$, expands adiabatically to $1.75$ times its initial volume. What are the final pressure and temperature for the gas? (Assume no molecular vibration.)

(II) A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end. The copper end is placed in a furnace maintained at a constant temperature of $225^{\circ} \mathrm{C}$. The aluminum end is placed in an ice bath held at constant temperature of $0.0^{\circ} \mathrm{C}$. Calculate the temperature at the point where the two rods are joined.

A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of 5.00 atm and a volume of 12.0 L to a final volume of 30.0 L. (a) What is the final pressure of the gas? (b) What are the initial and final temperatures? Find (c) Q (d) $\Delta E_{\text {int}}$ and (e) W for the gas during this process.

A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of $5.00 \mathrm{~atm}$ and a volume of $12.0 \mathrm{~L}$ to a final volume of $30.0 \mathrm{~L}$. Find $\Delta E_{\text {int }}$ for the gas during this process.

A 2.00-mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of $5.00 \mathrm{~atm}$ and a volume of $12.0 \mathrm{~L}$ to a final volume of $30.0 \mathrm{~L}$. Find $W$ for the gas during this process.

A 2.00-mole sample of $\mathrm{N}_2$ gas at $0^{\circ} \mathrm{C}$ is heated to $150^{\circ} \mathrm{C}$ at constant pressure $(1.00 \mathrm{~atm})$. Determine the change in internal energy.

A $2.00\text{-mole}$ sample of $\mathrm{N}_2$ gas at $0^{\circ} \mathrm{C}$ is heated to $150^{\circ} \mathrm{C}$ at constant pressure $(1.00 \mathrm{~atm})$. Determine the heat added to it.

A 2.00-mole sample of $\mathrm{N}_2$ gas at $0^{\circ} \mathrm{C}$ is heated to $150^{\circ} \mathrm{C}$ at constant pressure $(1.00 \mathrm{~atm})$. Determine the work the gas does.

Consider a parcel of air moving to a different altitude $y$ in the Earth's atmosphere (Fig. 19-34). As the parcel changes altitude it acquires the pressure $P$ of the surrounding air. From Eq. 13-4 we have

$$\frac{d P}{d y}=-\rho g$$

where $\rho$ is the parcel's altitude-dependent mass density.

During this motion, the parcel's volume will change and, because air is a poor heat conductor, we assume this expansion or contraction will take place adiabatically. Starting with Eq. 19-16, $P V^\gamma=$ constant, show that for an ideal gas undergoing an adiabatic process, $P^{1-\gamma} T^\gamma=$ constant. Then show that the parcel's pressure and temperature are related by

$$(1-\gamma) \frac{d P}{d y}+\gamma \frac{P}{T} \frac{d T}{d y}=0$$

and thus

$$(1-\gamma)(-\rho g)+\gamma \frac{P}{T} \frac{d T}{d y}=0 .$$

The specific heat at constant volume of a particular gas is $0.182 \mathrm{kcal} / \mathrm{kg} \cdot \mathrm{K}$ at room temperature, and its molecular mass is 34 . What is its specific heat at constant pressure?

In the p-V diagram of the given figure, the gas does 5 J of work when taken along isotherm ab and 4 J when taken along adiabat $b c$. What is the change in the internal energy of the gas when it is taken along the straight path from $a$ to $c$ ?