The gas equation for n moles of a real gas is `(P+(a)/(V^(2)))(V-b) = nRT` where P is the pressure, V is the volume, T is the absolute temperature, R is the molar gas constant and a, b are arbitrary constants. Which of the following have the same dimensions as those of PV ?

Options

A. nRT
B. `a//V`
C. Pb
D. `ab//V^(2)`

Answer

To solve the question, we need to analyze the given gas equation and determine which expressions have the same dimensions as \( PV \).

### Step-by-Step Solution:

1. **Understanding the Gas Equation**:
The gas equation given is:
\[ (P + \frac{a}{V^2})(V - b) = nRT \]
Here, \( P \) is pressure, \( V \) is volume, \( T \) is absolute temperature, \( R \) is the molar gas constant, and \( a \) and \( b \) are constants.

2. **Identifying Dimensions**:
- The dimension of pressure \( [P] \) is given by:
\[ [P] = \frac{M}{L \cdot T^2} \]
- The dimension of volume \( [V] \) is:
\[ [V] = L^3 \]
- The dimension of the gas constant \( R \) is:
\[ [R] = \frac{M}{L \cdot T^2 \cdot N} \]
- The dimension of \( n \) (number of moles) is:
\[ [n] = N \]
- The dimension of temperature \( [T] \) is:
\[ [T] = \Theta \]

3. **Finding Dimensions of \( nRT \)**:
The dimensions of \( nRT \) can be calculated as follows:
\[ [nRT] = [n] \cdot [R] \cdot [T] = N \cdot \frac{M}{L \cdot T^2 \cdot N} \cdot \Theta = \frac{M \cdot \Theta}{L \cdot T^2} \]

4. **Finding Dimensions of \( PV \)**:
The dimensions of \( PV \) are:
\[ [PV] = [P] \cdot [V] = \left(\frac{M}{L \cdot T^2}\right) \cdot (L^3) = \frac{M \cdot L^2}{T^2} \]

5. **Equating Dimensions**:
From the equation \( (P + \frac{a}{V^2})(V - b) = nRT \), we can see that the left-hand side must have the same dimensions as the right-hand side. Thus, we can conclude:
\[ \text{Dimensions of } (P + \frac{a}{V^2})(V - b) = \frac{M \cdot L^2}{T^2} \]

6. **Analyzing Each Option**:
- **Option A**: \( nRT \) has dimensions \( \frac{M \cdot L^2}{T^2} \) (Correct).
- **Option B**: \( \frac{a}{V^2} \) must also have dimensions of pressure, thus when multiplied by \( V \) gives \( PV \) (Correct).
- **Option C**: \( P \cdot (V - b) \) also results in \( PV \) (Correct).
- **Option D**: \( \frac{a}{V^2} \cdot V \cdot b \) gives \( PV \) (Correct).

### Conclusion:
All four options have the same dimensions as \( PV \).