Using the combined gas law below, 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 temperature.
[tex]\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}[/tex]
[tex]T_2=\frac{A}{B}[/tex]
Numerator $(A)$:
P_2 V_2 T_1
Denominator $(B)$:
P_1 V_1
Answer (2)
To solve for T 2 in the combined gas law, rearranging the equation leads to T 2 = P 1 V 1 P 2 V 2 T 1 . Therefore, the numerator ( A ) is T 1 P 2 V 2 and the denominator ( B ) is P 1 V 1 . This rearrangement allows us to understand how temperature varies with pressure and volume changes.
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Start with the combined gas law equation: T 1 P 1 V 1 = T 2 P 2 V 2 .
Multiply both sides by T 2 and T 1 to get T 2 P 1 V 1 = T 1 P 2 V 2 .
Divide both sides by P 1 V 1 to isolate T 2 : T 2 = P 1 V 1 P 2 V 2 T 1 .
Identify the numerator as P 2 V 2 T 1 and the denominator as P 1 V 1 , so the final answer is T 2 = P 1 V 1 P 2 V 2 T 1 .
Explanation
Understanding the Problem We are given the combined gas law: T 1 P 1 V 1 = T 2 P 2 V 2 Our goal is to rearrange this equation to solve for T 2 , and then identify the numerator (A) and denominator (B) in the rearranged equation T 2 = B A .
Multiplying by T 2 To isolate T 2 , we can start by multiplying both sides of the equation by T 2 : T 2 ⋅ T 1 P 1 V 1 = T 2 ⋅ T 2 P 2 V 2 This simplifies to: T 1 T 2 P 1 V 1 = P 2 V 2
Multiplying by T 1 Next, we multiply both sides of the equation by T 1 to get rid of the denominator on the left side: T 1 ⋅ T 1 T 2 P 1 V 1 = T 1 ⋅ P 2 V 2 This simplifies to: T 2 P 1 V 1 = T 1 P 2 V 2
Dividing by P 1 V 1 Now, to isolate T 2 , we divide both sides of the equation by P 1 V 1 : P 1 V 1 T 2 P 1 V 1 = P 1 V 1 T 1 P 2 V 2 This simplifies to: T 2 = P 1 V 1 T 1 P 2 V 2
Identifying Numerator and Denominator From the rearranged equation, we can identify the numerator (A) and the denominator (B): Numerator (A): P 2 V 2 T 1 Denominator (B): P 1 V 1
Final Answer Therefore, when the combined gas law is rearranged to solve for T 2 , the numerator (A) is P 2 V 2 T 1 and the denominator (B) is P 1 V 1 .
Examples
The combined gas law is useful in scenarios where you need to predict how the volume, pressure, or temperature of a gas will change. For example, if you have a balloon filled with air and you move it from a cold room to a warm room, you can use the combined gas law to predict how the volume of the balloon will change, assuming the pressure remains constant. This is crucial in many engineering applications, such as designing engines or predicting weather patterns.
