A movable piston having a mass of 8.00 kg and a cross-sectional area of 5.00 cm2 traps 0.200 moles of an ideal gas in a vertical cylinder. If the piston slides without friction in the cylinder, how much work is done on the gas when its temperature is increased from 18°C to 308°C?
Well, let's break it down, shall we? First, we need to find the initial pressure of the gas. We can use the Ideal Gas Law here, which states that PV=nRT.
Given:
n = 0.200 moles
R = 8.314 J/(mol·K)
We can rearrange the equation to find the initial pressure:
P = (nRT)/(V)
Now, we don't have the volume (V) of the gas, but we do have the cross-sectional area of the piston (A) and the mass (m) of the piston. So, we can calculate the initial volume of the gas using V = (A)(h), where h is the height the piston moves.
Given:
A = 5.00 cm^2
m = 8.00 kg
g = 9.8 m/s^2
We can find the height (h) using the weight of the piston:
F_gravity = m * g
F_gravity = P * A
m * g = P * A
h = (m * g) / (P * A)
Now that we have the initial volume (V) of the gas, we can substitute it back into the Ideal Gas Law formula to find the initial pressure (P). We'll use the initial temperature (T1), which is 18°C, converted to Kelvin (T = T1 + 273.15).
With the initial pressure (P1) calculated, we can now find the final pressure (P2) using the final temperature (T2), which is 308°C, converted to Kelvin (T = T2 + 273.15).
Now, let's find the work done on the gas using the formula:
Work = (P2 - P1) * V
Where P2 is the final pressure, P1 is the initial pressure, and V is the volume of the gas (A * h).
Do the math, and you'll find your answer. But hey, if you need any more help, just ask! I'm here to clown around and help you out!
To calculate the work done on the gas, we can use the formula:
Work = Pressure * Change in Volume
Since the piston slides without friction, we can assume that the pressure on both sides of the piston is the same. Moreover, since the piston slides vertically, the force acting on the piston due to the pressure is equal to the weight of the piston.
First, let's calculate the weight of the piston:
Weight = Mass * Acceleration due to gravity
Weight = 8.00 kg * 9.81 m/s^2
Weight = 78.48 N
Since the area of the piston is given in cm^2, we need to convert it to m^2:
Area = 5.00 cm^2 = (5.00 * 10^-4) m^2
Now, let's calculate the pressure on the gas:
Pressure = Force / Area
Pressure = Weight / Area
Pressure = 78.48 N / (5.00 * 10^-4) m^2
Pressure = 1.57 * 10^5 Pa
Next, we need to calculate the change in volume of the gas. We can use the ideal gas law to do that:
PV = nRT,
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
Let's calculate the initial and final volumes using the ideal gas law.
Initial volume:
P1V1 = nRT1,
V1 = nRT1 / P1,
where T1 = 18°C = (18 + 273) K
Final volume:
P2V2 = nRT2,
V2 = nRT2 / P2,
where T2 = 308°C = (308 + 273) K
Now that we have the initial and final volumes, we can calculate the change in volume:
Change in Volume = V2 - V1
Finally, we can calculate the work done on the gas using the formula:
Work = Pressure * Change in Volume
By plugging in the values, we can calculate the work done on the gas when its temperature is increased.