When 1.5 mol of a gas is heated at constant volume from 300 K to 350 K and the heat supplied to the gas is 750 J then the correct option is

Options

A. `q = DeltaU = 750 J, w = 0`
B. `q = w = 750 J, DeltaU = 0`
C. `q = w = 750 J, DeltaU = -750 J`
D. `q = 750 J, w = DeltaU = -750 J`

Answer

To solve the problem, we need to determine the specific heat capacity of the gas using the information provided. We can use the formula for heat transfer at constant volume:

\[ Q = nC_v \Delta T \]

Where:
- \( Q \) = heat supplied (in joules)
- \( n \) = number of moles of the gas
- \( C_v \) = molar specific heat capacity at constant volume (in J/mol·K)
- \( \Delta T \) = change in temperature (in K)

### Step 1: Identify the given values
From the problem statement, we have:
- \( n = 1.5 \, \text{mol} \)
- \( Q = 750 \, \text{J} \)
- Initial temperature \( T_1 = 300 \, \text{K} \)
- Final temperature \( T_2 = 350 \, \text{K} \)

### Step 2: Calculate the change in temperature (\( \Delta T \))
\[ \Delta T = T_2 - T_1 = 350 \, \text{K} - 300 \, \text{K} = 50 \, \text{K} \]

### Step 3: Rearrange the heat transfer formula to solve for \( C_v \)
\[ C_v = \frac{Q}{n \Delta T} \]

### Step 4: Substitute the known values into the equation
\[ C_v = \frac{750 \, \text{J}}{1.5 \, \text{mol} \times 50 \, \text{K}} \]

### Step 5: Calculate \( C_v \)
\[ C_v = \frac{750 \, \text{J}}{75 \, \text{mol·K}} = 10 \, \text{J/mol·K} \]

### Conclusion
The molar specific heat capacity \( C_v \) of the gas is \( 10 \, \text{J/mol·K} \).