The ratio a/b (the terms used in van der Waals' equation) has the unit .

Options

A. atm litre `mol^(-1)`
B. atm `dm^3 mol^(-1)`
C. dyne cm `mol^(-1)`
D. All of these

Answer

To determine the unit of the ratio \( \frac{a}{b} \) in the van der Waals equation, we need to analyze the units of \( a \) and \( b \) separately.

### Step 1: Identify the units of \( a \)
The term \( a \) in the van der Waals equation is associated with the attractive forces between particles and has the following units:
- \( a \) is expressed in units of pressure multiplied by volume squared per mole squared.
- In SI units, this can be represented as:
\[ a \text{ (units)} = \text{atm} \cdot \text{L}^2/\text{mol}^2 \]
or equivalently,
\[ a \text{ (units)} = 8 \text{ m} \cdot \text{dm}^6/\text{mol}^2 \]
or in CGS units,
\[ a \text{ (units)} = \text{dyne} \cdot \text{cm}^4/\text{mol}^2 \]

### Step 2: Identify the units of \( b \)
The term \( b \) represents the volume occupied by one mole of the gas and has the following units:
- \( b \) is expressed in units of volume per mole.
- In SI units, this can be represented as:
\[ b \text{ (units)} = \text{dm}^3/\text{mol} \]
or equivalently,
\[ b \text{ (units)} = \text{cm}^3/\text{mol} \]

### Step 3: Calculate the ratio \( \frac{a}{b} \)
Now, we can find the units of the ratio \( \frac{a}{b} \):
\[ \frac{a}{b} = \frac{\text{atm} \cdot \text{L}^2/\text{mol}^2}{\text{L}^3/\text{mol}} = \frac{\text{atm} \cdot \text{L}^2}{\text{L}^3} \cdot \frac{\text{mol}}{\text{mol}^2} = \frac{\text{atm}}{\text{L}} \cdot \frac{1}{\text{mol}} = \frac{\text{atm}}{\text{L} \cdot \text{mol}} \]
This can also be expressed in other units, such as:
\[ \frac{a}{b} = \frac{\text{dyne} \cdot \text{cm}^4/\text{mol}^2}{\text{cm}^3/\text{mol}} = \frac{\text{dyne} \cdot \text{cm}^4}{\text{cm}^3} \cdot \frac{\text{mol}}{\text{mol}^2} = \frac{\text{dyne}}{\text{cm}} \cdot \frac{1}{\text{mol}} = \frac{\text{dyne}}{\text{cm} \cdot \text{mol}} \]

### Conclusion
Thus, the ratio \( \frac{a}{b} \) can be expressed in various units, including:
- \( \frac{\text{atm}}{\text{L} \cdot \text{mol}} \)
- \( \frac{\text{dyne}}{\text{cm} \cdot \text{mol}} \)

### Final Answer
The ratio \( \frac{a}{b} \) has the units of \( \frac{\text{atm}}{\text{L} \cdot \text{mol}} \), \( \frac{\text{dyne}}{\text{cm} \cdot \text{mol}} \), and other equivalent forms.

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