The quantity of heat required to raise one mole for a monoatomic gas at constant volume through one degree celsius is

Options

A. R
B. `(3R)/(2)`
C. `(5R)/(2)`
D. `(7R)/(2)`

Explanation

To find the quantity of heat required to raise the temperature of one mole of a monoatomic gas at constant volume through one degree Celsius, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Specific Heat at Constant Volume (Cv)**:
The specific heat at constant volume (Cv) is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius per mole. For a monoatomic gas, the specific heat at constant volume is given by the formula:
\[ C_v = \frac{3}{2} R \]
where \( R \) is the universal gas constant.

2. **Identify the Variables**:
- For this problem, we have:
- \( n = 1 \) mole (since we are considering one mole of gas)
- \( dT = 1 \) degree Celsius (the temperature change we are considering)

3. **Use the Heat Transfer Formula**:
The amount of heat (\( dq \)) required to raise the temperature of a substance can be calculated using the formula:
\[ dq = n C_v dT \]

4. **Substitute the Known Values**:
Now, substituting the known values into the formula:
\[ dq = 1 \cdot \left(\frac{3}{2} R\right) \cdot 1 \]
This simplifies to:
\[ dq = \frac{3}{2} R \]

5. **Conclusion**:
Therefore, the quantity of heat required to raise the temperature of one mole of a monoatomic gas at constant volume through one degree Celsius is:
\[ dq = \frac{3}{2} R \]

### Final Answer:
The quantity of heat required is \( \frac{3}{2} R \).

---