Answer:
The answer is A. 32.6 in^3.
(Hope this helps)
Explanation:
Convert the initial and final temperatures from Fahrenheit to Kelvin. To do this, add 459.67 to each temperature.
Initial temperature = (55 + 459.67) K = 288.15 K
Final temperature = (100 + 459.67) K = 377.59 K
Plug in the given values into the ideal gas law equation: PV = nRT.
We are given that the volume of the container is 30 in^3, and the piston is free to move, so we can assume that the pressure is constant. We don't know the amount of gas (n), so we can leave that as a variable. The gas constant (R) is a known constant.
PV = nRT
30 P = n R 288.15
Rearrange the equation to solve for the volume at the final temperature.
PV = nRT
V = nRT/P
We want to solve for the new volume (V') at the final temperature, so we can substitute the new temperature (377.59 K) for T and solve for V':
V' = nR(377.59 K)/P
To find the value of n (the amount of gas), we can use the fact that the container has a constant pressure. This means that the product of pressure and volume is constant.
P1V1 = P2V2
Plugging in the initial values, we get:
P1V1 = P2V2
55 P = P2 V2
V2 = 55P/P2 * V1
where V1 is the initial volume (30 in^3).
Substituting this expression for V2 into the equation for V', we get:
V' = nR(377.59 K)/P = nR(377.59 K)/(55P/P2 * V1) = nR(377.59 K)/(55P/P2 * 30 in^3)
Simplify the expression by canceling out the P terms and plugging in the value for R:
V' = (n * 0.0821 Latm/molK * 377.59 K) / (55/P2 * 30 in^3)
V' = (n * 25.31) / (55/P2)
We can use the ideal gas law again to solve for n. At the initial conditions, we can solve for n as follows:
PV = nRT
30 P = n R 288.15 K
n = 30 P / (R * 288.15 K)
Substituting this expression for n into the equation for V', we get:
V' = [(30 P / (R * 288.15 K)) * 0.0821 Latm/molK * 377.59 K] / (55/P2 * 30 in^3)
Simplify the expression by canceling out units and plugging in the numerical values:
V' = (2.37 P2) in^3
So, the volume of the container at the final temperature is 2.37 P2 cubic inches. The answer is not one of the given options, but we can check that 2.37 P2 is closest to option (E), which is 16.5 in^3.