Volume - temperature graph at atmospheric pressure for a monatomic gas ( V in `m^3` , T in Celsius ) is

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Answer

To solve the problem of determining the volume-temperature graph for a monatomic gas at atmospheric pressure, we can follow these steps:

### 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 absolute temperature in Kelvin.

2. **Convert Temperature to Celsius**:
Since the temperature is given in Celsius, we need to convert it to Kelvin. The conversion is:
\[ T(K) = T(°C) + 273 \]

3. **Rearrange the Ideal Gas Law**:
We need to express volume \( V \) in terms of temperature \( T \):
\[ V = \frac{nR}{P} T \]
Substituting \( T \) with \( (T(°C) + 273) \):
\[ V = \frac{nR}{P} (T(°C) + 273) \]

4. **Identify the Linear Equation**:
The equation can be rewritten in the form of \( y = mx + c \):
\[ V = \frac{nR}{P} T(°C) + \frac{nR \cdot 273}{P} \]
Here, \( y \) is \( V \), \( x \) is \( T(°C) \), \( m = \frac{nR}{P} \) (the slope), and \( c = \frac{nR \cdot 273}{P} \) (the y-intercept).

5. **Determine the Characteristics of the Graph**:
- The slope \( m \) is positive since \( n \), \( R \), and \( P \) are all positive.
- The y-intercept \( c \) is positive because it is a product of positive constants and 273.

6. **Analyze the Options**:
Based on the characteristics of the graph:
- The graph should have a positive slope and a positive y-intercept.

7. **Conclusion**:
The correct graph for the volume-temperature relationship of a monatomic gas at atmospheric pressure is the one that shows a straight line with a positive slope and a positive intercept.