At a given volume and temperature the pressure of a gas

Options

A. varies inversely as its mass
B. varies inversely as the square of its mass
C. varies linearly as its mass
D. is independent of its mass

Answer

To solve the question regarding the relationship between the pressure of a gas, its volume, temperature, and mass, 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 of the gas, \( R \) is the universal gas constant, and \( T \) is the temperature.

**Hint**: Recall that the ideal gas law relates pressure, volume, temperature, and the amount of gas.

2. **Identify Constants**:
In this problem, we are given that the volume \( V \) and temperature \( T \) are constant. This means that \( R \) (the gas constant) is also constant.

**Hint**: When certain variables are held constant, it simplifies the relationship between the remaining variables.

3. **Rearranging the Ideal Gas Law**:
Since \( V \) and \( T \) are constant, we can rearrange the ideal gas law to express pressure in terms of the number of moles:
\[ P = \frac{nRT}{V} \]
Since \( R \) and \( V \) are constants, we can write:
\[ P \propto n \]

**Hint**: Proportionality means that if one variable increases, the other variable will increase or decrease in a predictable way.

4. **Relate Moles to Mass**:
The number of moles \( n \) can be expressed in terms of mass \( m \) and the molecular mass \( M \) of the gas:
\[ n = \frac{m}{M} \]
Substituting this into the pressure equation gives:
\[ P \propto \frac{m}{M} \]

**Hint**: Remember that the number of moles is directly related to the mass of the gas divided by its molecular mass.

5. **Conclude the Relationship**:
Since \( M \) (molecular mass) is a constant for a given gas, we can conclude that:
\[ P \propto m \]
This means that the pressure of the gas is directly proportional to its mass when the volume and temperature are held constant.

**Hint**: Direct proportionality indicates that if the mass increases, the pressure will also increase linearly.

6. **Evaluate the Options**:
Based on our findings, we can evaluate the provided options:
- The pressure varies universally as its mass: Incorrect.
- The pressure varies inversely as the square of its mass: Incorrect.
- The pressure varies linearly as its mass: Correct.
- The pressure is independent of its mass: Incorrect.

**Hint**: Check each option against your derived relationship to determine its validity.

### Final Answer:
At a given volume and temperature, the pressure of the gas varies linearly as its mass.