The equation PaVbTc= constant reduces to Charles Law if

Analysis

The question asks us to identify the values of the exponents $a, b,$ and $c$ in the general equation $P^a V^b T^c = \text{constant}$ that would transform it into Charles's Law.

Concept: Charles's Law

Charles's Law states that for a fixed mass of an ideal gas at constant pressure, the volume ($V$) is directly proportional to its absolute temperature ($T$).

Mathematically, this is expressed as:

Or, written in the form $V \cdot T^{-1} = \text{constant}$.

Step-by-Step Solution

1. Compare the equations: We are given the general form: $P^a V^b T^c = \text{constant}$. We want to reach the form: $V^1 T^{-1} = \text{constant}$.

2. Match the variables: • To eliminate pressure ($P$), the exponent $a$ must be 0 (since $P^0 = 1$). • To match the volume ($V$), the exponent $b$ must be 1. • To match the temperature ($T$), the exponent $c$ must be -1 (since $T^{-1} = \frac{1}{T}$).

3. Verify: Substituting $a=0, b=1, c=-1$ into the original equation: $P^0 \cdot V^1 \cdot T^{-1} = \text{constant}$ $1 \cdot V \cdot \frac{1}{T} = \text{constant}$ $\frac{V}{T} = \text{constant}$ This matches the definition of Charles's Law perfectly.

Conclusion

Comparing our findings ($a=0, b=1, c=-1$) with the given options, we see that it corresponds to option D.

Exam Tip

When dealing with gas law equations in JAMB, remember the "Constant" rule:

• Boyle’s Law: Temperature is constant ($PV = k$) $\rightarrow a=1, b=1, c=0$.
• Charles’s Law: Pressure is constant ($V/T = k$) $\rightarrow a=0, b=1, c=-1$.
• Pressure Law: Volume is constant ($P/T = k$) $\rightarrow a=1, b=0, c=-1$.