When two moles of a gas is heated from `O^(0)` to `10^(0)C` at constant volume, its internal enernal changes by `420J` . The moles specifie heat of the gas at constant volume

Options

A. `5.75 J K^(-1) "mole"^(-1)`
B. `10.55 J K^(-1) "mole"^(-1)`
C. `21 J K^(-1) "mole"^(-1)`
D. `42 J K^(-1) "mole"^(-1)`

Explanation

To find the molar specific heat of the gas at constant volume (Cv), we can follow these steps:

### Step 1: Understand the relationship between internal energy change and specific heat
The change in internal energy (ΔU) for an ideal gas at constant volume is given by the formula:
\[ \Delta U = n C_v \Delta T \]
where:
- \( \Delta U \) = change in internal energy
- \( n \) = number of moles of gas
- \( C_v \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

### Step 2: Identify the known values
From the problem, we know:
- \( \Delta U = 420 \, \text{J} \)
- \( n = 2 \, \text{moles} \)
- Initial temperature \( T_i = 0^\circ C = 273 \, \text{K} \)
- Final temperature \( T_f = 10^\circ C = 283 \, \text{K} \)

### Step 3: Calculate the change in temperature (ΔT)
\[ \Delta T = T_f - T_i = 283 \, \text{K} - 273 \, \text{K} = 10 \, \text{K} \]

### Step 4: Substitute the known values into the internal energy equation
Now we can substitute the known values into the internal energy equation:
\[ 420 \, \text{J} = 2 \, \text{moles} \times C_v \times 10 \, \text{K} \]

### Step 5: Solve for Cv
Rearranging the equation to solve for \( C_v \):
\[ C_v = \frac{420 \, \text{J}}{2 \, \text{moles} \times 10 \, \text{K}} = \frac{420}{20} = 21 \, \text{J/mol·K} \]

### Step 6: Conclusion
Thus, the molar specific heat of the gas at constant volume \( C_v \) is:
\[ C_v = 21 \, \text{J/mol·K} \]