Question

a cylinder with a moveable piston contains 4.78 l of a gas at 417 mm hg and 497 k. the pressure increases to 576 mm hg and temperature decreases to 386 k. what is the new volume of the cylinder?

Explanation:

Step1: Recall the combined gas law formula

The combined gas law is given by $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$, where $P_1, V_1, T_1$ are the initial pressure, volume, temperature and $P_2, V_2, T_2$ are the final pressure, volume, temperature. We need to solve for $V_2$. Rearranging the formula for $V_2$ gives $V_2=\frac{P_1V_1T_2}{P_2T_1}$.

Step2: Identify the given values

We have $P_1 = 417\space mm\space Hg$, $V_1 = 4.78\space L$, $T_1 = 497\space K$, $P_2 = 576\space mm\space Hg$, $T_2 = 386\space K$.

Step3: Substitute the values into the formula

Substitute the values into $V_2=\frac{P_1V_1T_2}{P_2T_1}$:

$$ V_2=\frac{417\space mm\space Hg\times4.78\space L\times386\space K}{576\space mm\space Hg\times497\space K} $$

Step4: Calculate the numerator and the denominator

First, calculate the numerator: $417\times4.78\times386 = 417\times4.78 = 2003.06$; $2003.06\times386 = 2003.06\times300+2003.06\times80 + 2003.06\times6=600918+160244.8+12018.36 = 773181.16$

Then, calculate the denominator: $576\times497=576\times(500 - 3)=576\times500-576\times3 = 288000 - 1728 = 286272$

Step5: Divide the numerator by the denominator

$V_2=\frac{773181.16}{286272}\approx2.69\space L$ (rounded to three significant figures)

Answer:

The new volume of the cylinder is approximately $\boldsymbol{2.69\space L}$