310 J of heat is required to rise the temperature of 2 moles of an ideal gas at constant pressure from `25^(@)C` to `35^(@)C`. The amount of heat required to raise the temperature of the gas through the same range at constant volume, is

Options

A. 384 J
B. 144 J
C. 276 J
D. 452 J

Explanation

To solve the problem of finding the amount of heat required to raise the temperature of 2 moles of an ideal gas at constant volume, we can follow these steps:

### Step 1: Understand the Given Information
We know that:
- Heat required at constant pressure (Q_p) = 310 J
- Number of moles (N) = 2 moles
- Initial temperature (T1) = 25°C
- Final temperature (T2) = 35°C

### Step 2: Calculate the Change in Temperature
The change in temperature (ΔT) can be calculated as:
\[ \Delta T = T2 - T1 = 35°C - 25°C = 10°C \]
(Note: The change in temperature is the same in Kelvin.)

### Step 3: Use the Formula for Heat at Constant Pressure
The formula for heat at constant pressure is:
\[ Q_p = N C_p \Delta T \]
Where:
- \(C_p\) = molar heat capacity at constant pressure

Substituting the known values:
\[ 310 J = 2 \, \text{moles} \times C_p \times 10 K \]

### Step 4: Solve for \(C_p\)
Rearranging the equation to find \(C_p\):
\[ C_p = \frac{310 J}{2 \times 10 K} = \frac{310 J}{20 K} = 15.5 \, \text{J/mol K} \]

### Step 5: Use the Relationship Between \(C_p\) and \(C_v\)
We know that:
\[ C_p - C_v = R \]
Where \(R\) is the ideal gas constant. For our calculations, we'll use \(R = 8.3 \, \text{J/mol K}\).

Now, we can find \(C_v\):
\[ C_v = C_p - R = 15.5 \, \text{J/mol K} - 8.3 \, \text{J/mol K} = 7.2 \, \text{J/mol K} \]

### Step 6: Calculate the Heat Required at Constant Volume
Now we can use the formula for heat at constant volume:
\[ Q_v = N C_v \Delta T \]
Substituting the known values:
\[ Q_v = 2 \, \text{moles} \times 7.2 \, \text{J/mol K} \times 10 K \]

### Step 7: Perform the Calculation
\[ Q_v = 2 \times 7.2 \times 10 = 144 J \]

### Final Answer
The amount of heat required to raise the temperature of the gas through the same range at constant volume is **144 J**.

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