The numerical value of `(RT)/(PV)` for a gas at critical condition is ....... Times of `(RT)/(PV)` at normal conditions

Options

A. 4
B. `3//8`
C. `8//3`
D. `1//4`

Explanation

To solve the problem, we need to determine the numerical value of \((RT)/(PV)\) for a gas at critical conditions in relation to its value at normal conditions.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
For one mole of gas (\(n = 1\)), this simplifies to:
\[ PV = RT \]

2. **Calculate \((RT)/(PV)\) at Normal Conditions**:
From the ideal gas law, we can express \((RT)/(PV)\) at normal conditions:
\[ \frac{RT}{PV} = 1 \]
This means that at normal conditions, the value of \((RT)/(PV)\) is 1.

3. **Consider Critical Conditions**:
At critical conditions, the relationship between pressure, volume, and temperature can be expressed as:
\[ P_c V_c = \frac{3}{8} RT_c \]
where \(P_c\) is the critical pressure, \(V_c\) is the critical volume, and \(T_c\) is the critical temperature.

4. **Calculate \((RT)/(PV)\) at Critical Conditions**:
We can rearrange the equation for critical conditions:
\[ PV = P_c V_c = \frac{3}{8} RT_c \]
Therefore, we can express \((RT)/(PV)\) at critical conditions as:
\[ \frac{RT}{PV} = \frac{RT}{\frac{3}{8} RT_c} \]
This simplifies to:
\[ \frac{RT}{PV} = \frac{8}{3} \cdot \frac{T}{T_c} \]

5. **Compare Values**:
Since we are interested in the ratio of \((RT)/(PV)\) at critical conditions to that at normal conditions:
\[ \text{At normal conditions: } \frac{RT}{PV} = 1 \]
\[ \text{At critical conditions: } \frac{RT}{PV} = \frac{8}{3} \cdot \frac{T}{T_c} \]
Thus, the numerical value of \((RT)/(PV)\) at critical conditions is:
\[ \frac{RT}{PV} = \frac{8}{3} \]

6. **Final Answer**:
Therefore, the numerical value of \((RT)/(PV)\) for a gas at critical conditions is \(\frac{8}{3}\) times that at normal conditions.

### Conclusion:
The answer is:
\[ \frac{8}{3} \]