Question

The temperature of 5 moles of an ideal gas increases by 20 K.
Given:
$C_{p} = 9 \text{ cal mol}^{-1}\text{K}^{-1}$, $R = 2 \text{ cal mol}^{-1}\text{K}^{-1}$
The change in internal energy of the gas is:

• A. 500 cal
• B. 700 cal
• C. 800 cal
• D. 900 cal

Hint:

A key property of an ideal gas is that its internal energy is a function of temperature only. Therefore, the formula $\Delta U = nC_v\Delta T$ is valid for *any* process (isobaric, isochoric, isothermal, adiabatic, etc.), not just for a constant volume process. You only need the initial and final temperatures.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to calculate the change in internal energy ($\Delta U$) for an ideal gas given the number of moles, the temperature change, the molar heat capacity at constant pressure ($C_p$), and the universal gas constant ($R$).
Step 2: Key Formula or Approach:
1. The change in internal energy for any process of an ideal gas depends only on the temperature change and is given by:
$\Delta U = n C_{v} \Delta T$
where $n$ is the number of moles, $C_v$ is the molar heat capacity at constant volume, and $\Delta T$ is the change in temperature.
2. For an ideal gas, the relationship between $C_p$ and $C_v$ is given by Mayer's relation:
$C_{p} - C_{v} = R$
Step 3: Detailed Explanation:
Given values:
Number of moles, $n = 5$ mol
Change in temperature, $\Delta T = 20$ K
$C_{p} = 9 \text{ cal mol}^{-1}\text{K}^{-1}$
$R = 2 \text{ cal mol}^{-1}\text{K}^{-1}$
First, we need to find the molar heat capacity at constant volume, $C_v$. Using Mayer's relation:
\[ C_{v} = C_{p} - R \] \[ C_{v} = 9 - 2 = 7 \text{ cal mol}^{-1}\text{K}^{-1} \] Now, we can calculate the change in internal energy, $\Delta U$:
\[ \Delta U = n C_{v} \Delta T \] \[ \Delta U = (5 \text{ mol}) \times (7 \text{ cal mol}^{-1}\text{K}^{-1}) \times (20 \text{ K}) \] \[ \Delta U = 35 \times 20 = 700 \text{ cal} \] Step 4: Final Answer:
The change in internal energy of the gas is 700 cal.