Which of the following statements is false?

Options

A. The product of pressure and volume of fixed amount of a gas is independent of temperature.
B. Molecules of different gasses have the same kinetic energy at a given temperature.
C. The gas equation is not valid at high pressure and low temperature.
D. The gas constant per molecule is known as Boltzmann's constant.

Explanation

To determine which statement is false, we will analyze each statement one by one based on our understanding of the gas laws and kinetic theory.

### Step 1: Analyze the First Statement
**Statement:** The product of pressure and volume of a fixed amount of gas is independent of temperature.

**Analysis:** According to the ideal gas law, \( PV = nRT \), where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature

From this equation, we can see that the product \( PV \) is directly proportional to the temperature \( T \). Therefore, if temperature changes, the product of pressure and volume will also change.

**Conclusion:** This statement is **false**.

### Step 2: Analyze the Second Statement
**Statement:** Molecules of different gases have the same kinetic energy at a given temperature.

**Analysis:** The average kinetic energy of gas molecules is given by the formula:
\[ KE = \frac{3}{2} kT \]
where \( k \) is Boltzmann's constant and \( T \) is the temperature. At a given temperature, all gas molecules, regardless of their type, will have the same average kinetic energy.

**Conclusion:** This statement is **true**.

### Step 3: Analyze the Third Statement
**Statement:** The gas equation is not valid at high pressure and low temperature.

**Analysis:** The ideal gas law assumes that gas molecules do not interact and occupy no volume. At high pressures, gas molecules are forced closer together, and at low temperatures, they have less kinetic energy, which can lead to interactions and condensation into liquids. Therefore, the ideal gas law does not hold true under these conditions.

**Conclusion:** This statement is **true**.

### Step 4: Analyze the Fourth Statement
**Statement:** The gas constant per molecule is known as the Boltzmann constant.

**Analysis:** The universal gas constant \( R \) is related to the Boltzmann constant \( k \) by the equation:
\[ R = N_A k \]
where \( N_A \) is Avogadro's number. Thus, the gas constant per molecule indeed corresponds to the Boltzmann constant.

**Conclusion:** This statement is **true**.

### Final Conclusion
After analyzing all four statements, we find that the only false statement is the first one.

**Correct Answer:** The false statement is the first one.

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