For an ideal gas which of the following graphs will not be straight line when all the other variables are held constant?

Options

A. P vs T
B. V vs T
C. P vs 1/V
D. n vs T

Answer

To determine which graph will not be a straight line for an ideal gas when all other variables are held constant, we can analyze each option based on the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Absolute temperature

Now, let's analyze each graph one by one:

### Step 1: Analyze P vs T Graph
- **Variables Held Constant**: Volume (V) and Number of Moles (n)
- **Relationship**: According to Gay-Lussac's Law, pressure is directly proportional to temperature when volume and number of moles are constant.
- **Conclusion**: This graph will be a straight line.

### Step 2: Analyze V vs T Graph
- **Variables Held Constant**: Pressure (P) and Number of Moles (n)
- **Relationship**: According to Charles' Law, volume is directly proportional to temperature when pressure and number of moles are constant.
- **Conclusion**: This graph will also be a straight line.

### Step 3: Analyze P vs 1/V Graph
- **Variables Held Constant**: Temperature (T) and Number of Moles (n)
- **Relationship**: According to Boyle's Law, pressure is inversely proportional to volume, which can be expressed as \( P \propto \frac{1}{V} \) when temperature and number of moles are constant.
- **Conclusion**: This graph will also be a straight line.

### Step 4: Analyze N vs T Graph
- **Variables Held Constant**: Pressure (P) and Volume (V)
- **Relationship**: From the ideal gas equation, if \( PV = nRT \) is rearranged to \( n = \frac{PV}{RT} \), we see that number of moles (n) is inversely proportional to temperature (T) when pressure and volume are constant.
- **Conclusion**: This graph will not be a straight line; it will be a hyperbola.

### Final Answer
The graph that will **not** be a straight line when all other variables are held constant is **N vs T** (option 4).

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