Heat capacity `(C_(V))` of an ideal gas is X KJ/mole/K. To rise its temperature from 298 K to 318 K, heat to be supplied per 10g gas will be (in KJ) [MW = 16]

Options

A. 16X
B. `6.25 X`
C. 32X
D. `12.5 X`

Explanation

To solve the problem of calculating the heat required to raise the temperature of 10 grams of an ideal gas with a given heat capacity, we can follow these steps:

### Step 1: Identify the given values
- Heat capacity at constant volume, \( C_V = X \) KJ/mole/K
- Initial temperature, \( T_1 = 298 \) K
- Final temperature, \( T_2 = 318 \) K
- Mass of the gas, \( m = 10 \) g
- Molar mass of the gas, \( MW = 16 \) g/mole

### Step 2: Calculate the change in temperature (\( \Delta T \))
\[ \Delta T = T_2 - T_1 = 318 \, \text{K} - 298 \, \text{K} = 20 \, \text{K} \]

### Step 3: Calculate the number of moles of the gas (\( N \))
To find the number of moles, use the formula:
\[ N = \frac{m}{MW} = \frac{10 \, \text{g}}{16 \, \text{g/mole}} = \frac{10}{16} = \frac{5}{8} \, \text{moles} \]

### Step 4: Use the formula for heat (\( Q \))
The heat supplied can be calculated using the formula:
\[ Q = N \cdot C_V \cdot \Delta T \]
Substituting the values we have:
\[ Q = \left( \frac{5}{8} \right) \cdot X \cdot 20 \]

### Step 5: Simplify the expression
\[ Q = \frac{5 \cdot 20}{8} \cdot X = \frac{100}{8} \cdot X = 12.5 \cdot X \, \text{KJ} \]

### Conclusion
The heat required to raise the temperature of 10 grams of the gas from 298 K to 318 K is:
\[ Q = 12.5 \, X \, \text{KJ} \]