Answered

$\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}$



Using the combined gas law above, identify the variables that would be in the numerator $(A)$ and denominator $(B)$ if you were to rearrange the gas law to solve for final pressure:

$P_2=\frac{A}{B}$



Numerator $(A)$:

A$P_1 V_1 T_2$
B$P_2 V_2 T_1$
C$P_1 V_1$



Denominator $(B)$:

A$P_1 V_1 T_2$
B$P_2 V_2 T_1$
C$T_1 V_2$
D$P_1 V_1$

Verified Answer :

  • The combined gas law is defined as \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
  • To isolate the final pressure \(P_2\), multiply by \(T_2\) and divide by \(V_2\), yielding \(P_2 = \frac{P_1 V_1 T_2}{T_1 V_2}\).
  • The numerator \(A\) is \(\boxed{P_1 V_1 T_2}\).
  • The denominator \(B\) is \(\boxed{T_1 V_2}\).

Explanation

  1. Analysis of the Combined Gas Law

The combined gas law relates the pressure, volume, and temperature of a gas under two different sets of conditions. The given formula is:
\[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]
where \(P_1, V_1, T_1\) are the initial pressure, volume, and temperature, and \(P_2, V_2, T_2\) are the final pressure, volume, and temperature. Our objective is to isolate \(P_2\) to express it in the form \(P_2 = \frac{A}{B}\).

  1. Solving for \(P_2\)

To isolate \(P_2\), we start with the original equation and multiply both sides by \(T_2\):
\[P_2 V_2 = \frac{P_1 V_1}{T_1} \times T_2\]
\[P_2 V_2 = \frac{P_1 V_1 T_2}{T_1}\]
Next, we divide both sides by \(V_2\) to leave \(P_2\) by itself on the left-hand side:
\[P_2 = \frac{P_1 V_1 T_2}{T_1 V_2}\]

  1. Identifying the Components \(A\) and \(B\)

Comparing our derived expression \(P_2 = \frac{P_1 V_1 T_2}{T_1 V_2}\) to the target form \(P_2 = \frac{A}{B}\), we identify the components:

  • The numerator \(A\) is \(P_1 V_1 T_2\).
  • The denominator \(B\) is \(T_1 V_2\).

Thus, the required variables in the numerator are \(P_1, V_1, T_2\), and the variables in the denominator are \(T_1, V_2\). Comparing these with the options:

  • Numerator \(A\): The correct choice is \(P_1 V_1 T_2\).
  • Denominator \(B\): The correct choice is \(T_1 V_2\).

Examples

The combined gas law is essential in fields like chemistry and thermodynamics for predicting how gas behavior changes under different conditions. For example, if a weather balloon rises into the atmosphere where the temperature decreases and the pressure changes, scientists use this law to calculate how the volume of the gas inside the balloon will expand or contract. By rearranging the formula as we did, they can quickly determine the final state of the gas given the changes in the environment.

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