- Start with the combined gas law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
- Perform cross-multiplication to get: \(P_1 V_1 T_2 = P_2 V_2 T_1\).
- Divide by \(P_1 V_1\) to isolate \(T_2\): \(T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}\).
- Identify the numerator as \(A = P_2 V_2 T_1\) and the denominator as \(B = P_1 V_1\).
- The final expression is \(\boxed{T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}}\).
Explanation
- Analyzing the Problem
The goal is to rearrange the combined gas law equation \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) to isolate the variable \(T_2\). The initial pressure, volume, and temperature are represented by \(P_1\), \(V_1\), and \(T_1\), and the final conditions are represented by \(P_2\), \(V_2\), and \(T_2\). By isolating \(T_2\), we can determine how the final temperature is related to the other variables.
- Cross-Multiplication
To solve for \(T_2\), we start with the original equation:
\[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]
We perform cross-multiplication to clear the fractions. By multiplying both sides by \(T_1 T_2\), we obtain:
\[P_1 V_1 T_2 = P_2 V_2 T_1\]
- Isolating the Target Variable
Now, we isolate the variable \(T_2\) by dividing both sides of the equation by the term \(P_1 V_1\):
\[T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}\]
This gives us the final temperature expressed in the form \(T_2 = \frac{A}{B}\). Comparing this expression to the form provided, we identify the numerator \(A\) and the denominator \(B\).
- Final Identification
Based on the derivation, we have:
Numerator \(A = P_2 V_2 T_1\)
Denominator \(B = P_1 V_1\)
Thus, the final rearranged equation is:
\[T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}\]
Comparing these with the provided options:
Numerator \((A)\) is \(P_2 V_2 T_1\).
Denominator \((B)\) is \(P_1 V_1\).
The solution for \(T_2\) is \(\boxed{T_2 = \frac{P_2 V_2 T_1}{P_1 V_1}}\).
Examples
The combined gas law is a fundamental principle in chemistry and thermodynamics. It is frequently used by researchers to predict how a gas will behave under changing environmental conditions. For instance, when designing a pressure vessel or understanding how the air inside a car tire changes as the temperature increases after a long drive, engineers use this law to calculate how pressure, volume, and temperature will adjust to reach a new equilibrium state. By mastering this algebraic rearrangement, you gain the ability to predict these changes accurately, which is essential for safety and efficiency in various industrial and scientific applications.