The slope of the graph between log P and log V at constant temperature for a given mass of gas is

Options

A. `+1`
B. `-1`
C. `1/T`
D. `1/n`

Answer

To find the slope of the graph between log P and log V at constant temperature for a given mass of gas, we can start from the ideal gas law and manipulate it step by step.

### Step-by-Step Solution:

1. **Start with 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 of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature.

2. **Take the Logarithm of Both Sides**:
Taking the logarithm of both sides gives:
\[ \log(PV) = \log(nRT) \]

3. **Use Logarithmic Properties**:
We can use the property of logarithms that states \(\log(ab) = \log a + \log b\):
\[ \log P + \log V = \log(nRT) \]

4. **Rearranging the Equation**:
Rearranging this equation gives:
\[ \log P = \log(nRT) - \log V \]
This can be rewritten as:
\[ \log P = -\log V + \log(nRT) \]

5. **Identify the Linear Form**:
This equation is in the form of \( y = mx + c \), where:
- \( y = \log P \)
- \( x = \log V \)
- \( m = -1 \) (the slope)
- \( c = \log(nRT) \) (the y-intercept)

6. **Conclusion**:
From the above analysis, we can conclude that the slope of the graph between log P and log V at constant temperature for a given mass of gas is:
\[ \text{slope} = -1 \]

### Final Answer:
The slope of the graph between log P and log V at constant temperature for a given mass of gas is \(-1\).