Solved on Mar 04, 2024

Find T2T_2 given t2P1V1T1=P2V2T2t_2 \cdot \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}.

STEP 1

Assumptions
1. The equation given is t2P1V1T1=P2V2T2t_{2} \cdot \frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}}.
2. We need to solve for T2T_{2}.
3. All variables except T2T_{2} are assumed to be known and non-zero to avoid division by zero.

STEP 2

Isolate the term containing T2T_{2} on one side of the equation.
t2P1V1T1=P2V2T2t_{2} \cdot \frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}}

STEP 3

Multiply both sides of the equation by T2T_{2} to remove the fraction on the right side.
T2t2P1V1T1=P2V2T_{2} \cdot t_{2} \cdot \frac{P_{1} V_{1}}{T_{1}} = P_{2} V_{2}

STEP 4

Divide both sides of the equation by t2P1V1T1t_{2} \cdot \frac{P_{1} V_{1}}{T_{1}} to isolate T2T_{2} on the left side.
T2=P2V2t2P1V1T1T_{2} = \frac{P_{2} V_{2}}{t_{2} \cdot \frac{P_{1} V_{1}}{T_{1}}}

STEP 5

Simplify the right side of the equation by inverting the denominator and multiplying.
T2=P2V2T1t2P1V1T_{2} = \frac{P_{2} V_{2} T_{1}}{t_{2} \cdot P_{1} V_{1}}

STEP 6

Now we have the expression for T2T_{2} in terms of the other variables.
T2=P2V2T1t2P1V1T_{2} = \frac{P_{2} V_{2} T_{1}}{t_{2} P_{1} V_{1}}
This is the solution for T2T_{2} in terms of P1P_{1}, V1V_{1}, T1T_{1}, P2P_{2}, V2V_{2}, and t2t_{2}.

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