Question:\(\mathrm{P = P_a + \frac{k}{V}(T - T_a)}\)The formula above is used to model the pressure of a gas in a...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{P = P_a + \frac{k}{V}(T - T_a)}\)
The formula above is used to model the pressure of a gas in a rigid container as temperature changes, where P is the pressure and T is the temperature of the gas, V is the volume of the container, and \(\mathrm{P_a}\), \(\mathrm{T_a}\), and k are constants. Which of the following expresses the temperature T in terms of P, V, \(\mathrm{P_a}\), \(\mathrm{T_a}\), and k?
1. INFER the approach needed
- Goal: Get T by itself on one side of the equation
- Strategy: Work backwards from the structure - T is inside parentheses, multiplied by a fraction, then added to \(\mathrm{P_a}\)
- Plan: Undo each operation in reverse order: subtract \(\mathrm{P_a}\), multiply by reciprocal, add \(\mathrm{T_a}\)
2. SIMPLIFY by removing \(\mathrm{P_a}\)
Starting equation: \(\mathrm{P = P_a + \frac{k}{V}(T - T_a)}\)
Subtract \(\mathrm{P_a}\) from both sides:
\(\mathrm{P - P_a = \frac{k}{V}(T - T_a)}\)
3. SIMPLIFY by clearing the fraction
The term \(\mathrm{(T - T_a)}\) is multiplied by \(\mathrm{\frac{k}{V}}\). To isolate it, multiply both sides by the reciprocal \(\mathrm{\frac{V}{k}}\):
\(\mathrm{\frac{V}{k}(P - P_a) = T - T_a}\)
4. SIMPLIFY by isolating T
Add \(\mathrm{T_a}\) to both sides:
\(\mathrm{T = T_a + \frac{V}{k}(P - P_a)}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students struggle with reciprocals and make the error of multiplying by \(\mathrm{\frac{k}{V}}\) instead of its reciprocal \(\mathrm{\frac{V}{k}}\) when trying to clear the fraction.
When they multiply both sides by \(\mathrm{\frac{k}{V}}\), they get:
\(\mathrm{\frac{k}{V}(P - P_a) = \frac{k^2}{V^2}(T - T_a)}\)
This makes the problem more complex rather than simpler, leading to confusion and potentially selecting Choice A if they somehow reverse-engineer incorrectly.
Second Most Common Error:
Poor INFER reasoning: Students attempt to distribute or rearrange terms without a clear strategy for isolating T, leading to sign errors or incorrect algebraic steps.
For example, they might incorrectly distribute to get \(\mathrm{P = P_a + \frac{kT}{V} - \frac{kT_a}{V}}\) and then struggle to collect T terms properly. This leads to confusion and guessing among the remaining choices.
The Bottom Line:
Success depends on recognizing that solving for a variable requires systematically "undoing" operations in reverse order, and being careful with reciprocals when clearing fractions.