One mole of an ideal monoatomic gas requires 207 J heat to raise the temperature by 10 K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same 10 K, the heat required is [Given the gas constant R = 8.3 J/ mol. K]

Options

A. 198.7 J
B. 29 J
C. 215.3 J
D. 124 J

Answer

To solve the problem, we need to find the heat required to raise the temperature of one mole of an ideal monoatomic gas by 10 K at constant volume, given that it requires 207 J at constant pressure.

### Step 1: Understand the relationship between heat, specific heat capacities, and temperature change
For an ideal gas, the heat added at constant pressure (Q) can be expressed as:
\[ Q = n C_p \Delta T \]
Where:
- \( n \) = number of moles (1 mole in this case)
- \( C_p \) = specific heat capacity at constant pressure
- \( \Delta T \) = change in temperature (10 K)

### Step 2: Calculate \( C_p \)
From the problem, we know:
\[ Q = 207 \, \text{J} \]
\[ n = 1 \, \text{mol} \]
\[ \Delta T = 10 \, \text{K} \]

Using the formula:
\[ 207 = 1 \cdot C_p \cdot 10 \]
We can solve for \( C_p \):
\[ C_p = \frac{207}{10} = 20.7 \, \text{J/(mol K)} \]

### Step 3: Relate \( C_p \) and \( C_v \)
For an ideal monoatomic gas, the relationship between \( C_p \) and \( C_v \) is given by:
\[ C_p = C_v + R \]
Where \( R \) is the gas constant (8.3 J/(mol K)). Rearranging gives:
\[ C_v = C_p - R \]

### Step 4: Calculate \( C_v \)
Substituting the values we have:
\[ C_v = 20.7 - 8.3 = 12.4 \, \text{J/(mol K)} \]

### Step 5: Calculate the heat required at constant volume
Now, we can find the heat required to raise the temperature by 10 K at constant volume using the formula:
\[ Q = n C_v \Delta T \]
Substituting the values:
\[ Q = 1 \cdot 12.4 \cdot 10 \]
\[ Q = 124 \, \text{J} \]

### Conclusion
The heat required to raise the temperature of the gas by 10 K at constant volume is **124 J**.

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