Consider the figure below where a cylinder of an ideal gas is closed by an 16 kg movable piston of area = 60 cm2. When the gas is heated from 30 0C to 100 0C the piston rises by 20 cm. The piston is then fastened in place and the gas is cooled back to its original temperature of 30 0C. Find the difference in heat between the heating process (∆Q1 ) and the cooling proecess (∆Q2 ); ∆Q1 -∆Q2 . The picture has a piston on gas in a container(like a syringe).
To find the difference in heat between the heating process (∆Q1) and the cooling process (∆Q2), we need to apply the first law of thermodynamics: ΔQ = ΔU + ΔW, where ΔQ is the heat absorbed or released, ΔU is the change in internal energy of the gas, and ΔW is the work done on or by the gas.
In this case, the gas is an ideal gas, so the internal energy change is given by the formula ΔU = nCvΔT, where n is the number of moles of the gas, Cv is the molar specific heat capacity at constant volume, and ΔT is the change in temperature.
First, let's find the number of moles of the gas. We can use the ideal gas law: PV = nRT. Rearranging the equation, we get n = PV/RT. Given that the gas is ideal, we can assume that P is constant, so n is proportional to V, the volume of the gas. Since the piston rises by 20 cm, the volume increases by ΔV = Δh × A, where Δh is the change in height of the piston and A is the cross-sectional area of the piston.
To find the work done by the gas, we use the formula ΔW = PΔV, where P is the pressure of the gas. The pressure is given by P = F/A, where F is the force exerted by the piston. The force can be calculated by F = mg, where m is the mass of the piston and g is the acceleration due to gravity.
To calculate the change in temperature, we can use ΔT = T2 - T1, where T2 is the final temperature (100°C) and T1 is the initial temperature (30°C).
Now we have all the necessary values to calculate the difference in heat between the heating process (∆Q1) and the cooling process (∆Q2).
1. Calculate the number of moles of the gas:
n = PV/RT
2. Calculate the change in volume of the gas:
ΔV = Δh × A
3. Calculate the force exerted by the piston:
F = mg
4. Calculate the work done by the gas:
ΔW = PΔV
5. Calculate the change in internal energy of the gas:
ΔU = nCvΔT
6. Calculate the heat absorbed or released during heating (∆Q1):
∆Q1 = ΔU + ΔW
7. Let's assume that the cooling process is reversible, so ∆Q2 = -∆Q1.
8. Calculate the difference in heat between the heating and cooling processes:
∆Q1 - ∆Q2 = ∆Q1 - (-∆Q1) = 2∆Q1
By following these steps and plugging in the given values, you should be able to find the difference in heat between the heating process (∆Q1) and the cooling process (∆Q2) in terms of the provided values.
To find the difference in heat between the heating process (∆Q1) and the cooling process (∆Q2), we can use the equation:
∆Q = ∆U + W
Where ∆Q is the heat transferred to or from the gas, ∆U is the change in internal energy of the gas, and W is the work done on or by the gas.
Let's calculate each term step by step:
Step 1: Calculate the change in internal energy (∆U).
Since the cylinder is closed, the volume of the gas remains constant. Therefore, ∆U = 0 because there is no work being done on or by the gas.
Step 2: Calculate the work done on or by the gas (W).
The work done on or by the gas can be calculated using the formula:
W = F * d
Where F is the force and d is the distance the piston moves.
The force can be calculated using the equation:
F = P * A
Where P is the pressure and A is the area of the piston.
Given:
Mass of the piston = 16 kg
Area of the piston = 60 cm^2 = 0.006 m^2
Change in height of the piston = 20 cm = 0.20 m
We can find the pressure (P) using the formula:
P = F / A
We can find the force (F) using the formula:
F = m * g
Where m is the mass of the piston and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Finally, we can calculate the work done (W) using the formula:
W = F * d
Substituting the values, we get:
W = (m * g) * d
Step 3: Calculate the heat transferred (∆Q) for each process.
During the heating process, the gas absorbs heat (∆Q1).
During the cooling process, the gas releases heat (∆Q2).
We can calculate the heat transferred (∆Q) using the equation:
∆Q = ∆U + W
Since the change in internal energy (∆U) is zero, the equation simplifies to:
∆Q = W
Therefore, ∆Q1 = W1 and ∆Q2 = W2.
Step 4: Calculate the difference in heat between the two processes (∆Q1 - ∆Q2).
Let's calculate these step by step:
Step 1:
∆U = 0
Step 2:
F = m * g = 16 kg * 9.8 m/s^2 = 156.8 N
P = F / A = 156.8 N / 0.006 m^2 = 26,133.33 Pa
W1 = F * d = 156.8 N * 0.20 m = 31.36 J
Step 3:
∆Q1 = W1 = 31.36 J
∆Q2 = W2 (which we need to calculate)
Step 4:
∆Q1 - ∆Q2 = 31.36 J - ∆Q2
To find the value of ∆Q2, we need to calculate the work done during the cooling process. Since the piston is fastened in place, there is no change in height, and therefore no work is done (W2 = 0). Hence,
∆Q1 - ∆Q2 = 31.36 J - 0 J = 31.36 J
So, the difference in heat between the heating (∆Q1) and cooling (∆Q2) processes is 31.36 J.