At constant temperature if a graph plotted between logP and `log((1)/(V))`  has an intercept of unity then what will be the value of constant (k)

Options

A. 1
B. 10
C. 100
D. 1000

Answer

To solve the problem, we need to analyze the relationship between pressure (P) and volume (V) of an ideal gas at constant temperature and how it relates to logarithmic functions.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

2. **Rearranging the Equation**:
At constant temperature, we can rearrange the ideal gas law to express pressure in terms of volume:
\[ P = \frac{nRT}{V} \]
This shows that pressure \( P \) is inversely proportional to volume \( V \).

3. **Expressing the Relationship in Terms of a Constant**:
We can express this relationship as:
\[ PV = k \]
where \( k = nRT \) is a constant at constant temperature.

4. **Taking Logarithms**:
To analyze the relationship in logarithmic terms, we take the logarithm of both sides:
\[ \log P = \log k + \log \left(\frac{1}{V}\right) \]
This can be rewritten as:
\[ \log P = \log k - \log V \]

5. **Identifying the Intercept**:
The equation \( \log P = \log k + \log \left(\frac{1}{V}\right) \) resembles the equation of a straight line \( y = mx + c \), where:
- \( y \) is \( \log P \)
- \( x \) is \( \log \left(\frac{1}{V}\right) \)
- \( m \) is the slope
- \( c \) (the y-intercept) is \( \log k \)

6. **Using the Given Information**:
We are given that the intercept (c) is equal to 1. Therefore:
\[ \log k = 1 \]

7. **Solving for k**:
To find \( k \), we can exponentiate both sides:
\[ k = 10^1 = 10 \]

### Final Answer:
Thus, the value of the constant \( k \) is:
\[ \boxed{10} \]