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 step by step, we will analyze the relationship between pressure (P) and volume (V) of a gas at constant temperature and derive the value of the constant \( k \).

### Step 1: Understand the relationship between P and V
For a fixed amount of gas at constant temperature, we know from Boyle's Law that pressure (P) is inversely proportional to volume (V). This can be expressed mathematically as:
\[ P \propto \frac{1}{V} \]
This means that:
\[ P = k \cdot \frac{1}{V} \]
where \( k \) is a constant.

### Step 2: Take the logarithm of both sides
Now, we take the logarithm of both sides of the equation:
\[ \log P = \log k + \log \left(\frac{1}{V}\right) \]
Using the property of logarithms that states \( \log \left(\frac{1}{V}\right) = -\log V \), we can rewrite the equation as:
\[ \log P = \log k - \log V \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ \log P + \log V = \log k \]
or
\[ \log P = -\log V + \log k \]
This is in the form of a linear equation \( y = mx + c \), where:
- \( y = \log P \)
- \( m = -1 \) (the slope)
- \( x = \log V \)
- \( c = \log k \) (the y-intercept)

### Step 4: Analyze the graph
According to the problem, the graph plotted between \( \log P \) and \( \log \left(\frac{1}{V}\right) \) has an intercept of unity. This means:
\[ \log k = 1 \]

### Step 5: Solve for k
To find \( k \), we convert the logarithmic form back to its exponential form:
\[ k = 10^1 = 10 \]

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

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