Proof question related to the Ideal Gas Law

In summary, the pressure and volume of a gas are related according to: 𝑝 − 𝑝1 ∝ 𝑉 − 𝑉1
  • #1
Kajan thana
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A cylinder contains an initial volume V1 = 1m^^3 of a perfect gas at initial pressure p1 = 1 bar, confined by a piston that is held in place by a spring. The gas is heated until its volume is doubled and the final pressure is 5 bar. Assuming that the mass of the piston is negligible and that the initial force on the spring is zero, show that the pressure and volume V are related according to: 𝑝 − 𝑝1 ∝ 𝑉 − 𝑉1

I tried to work this backward so 𝑝 − 𝑝1 = k (𝑉 − 𝑉1) where k is the constant. After that, I don't know how to go about solving this question. Are we assuming the temperature remains constant in this question?
 
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  • #2
From freshman physics, we learned how do do an equilibrium force balance. So, if the area of the piston is A and the displacement of the piston (and spring) is x, what is your equilibrium force balance?
 
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  • #3
One observation is that the external pressure must also be ## p_1 ##. The rest is basically an ideal spring problem.
Note: They could also ask you to compute the final temperature w.r.t. the initial temperature, (assuming ideal gas law), and they could even ask you to compute the spring constant ## k ## in terms of the other parameters including the area).
 
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  • #4
Forces need to balance so it will follow as this pA = kx + P1A and we know that change of volume, V-V1 is proportional to x where x is the length of the spring that is getting compressed. With a few rearrangements, I should get the answer.
 
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  • #5
Note that you know the constant of proportionality here: ## V-V_1=Ax ##.
 
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  • #6
Charles Link said:
Note that you know the constant of proportionality here: ## V-V_1=Ax ##.
Thank you Charles and Chestermiller :angel::angel:
 
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  • #7
Let's see your final result=what do you get for ## C ## where ## p-p_1=C(V-V_1) ##?
 
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  • #8
Chestermiller said:
From freshman physics, we learned how do do an equilibrium force balance. So, if the area of the piston is A and the displacement of the piston (and spring) is x, what is your equilibrium force balance?

Charles Link said:
Let's see your final result=what do you get for ## C ## where ## p-p_1=C(V-V_1) ##?
Does C = k/A2 where k is the spring constant and A is area of the piston?
 
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  • #9
Kajan thana said:
Does C = k/A2 where k is the spring constant and A is area of the piston?
Thank you again.. you are a star..
 
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Related to Proof question related to the Ideal Gas Law

1. What is the Ideal Gas Law?

The Ideal Gas Law is a mathematical equation that describes the relationship between the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

2. What are the units of measurement for each variable in the Ideal Gas Law?

The units of measurement for the variables in the Ideal Gas Law are as follows: P is measured in atmospheres (atm), V is measured in liters (L), n is measured in moles (mol), R is measured in liters-atmospheres per mole-kelvin (L·atm/mol·K), and T is measured in kelvin (K).

3. How is the Ideal Gas Law derived?

The Ideal Gas Law is derived from the combination of three other gas laws: Boyle's Law, Charles's Law, and Avogadro's Law. These laws describe the relationships between pressure and volume, temperature and volume, and volume and number of moles, respectively. By combining these laws and using the universal gas constant, we can arrive at the Ideal Gas Law.

4. What is an ideal gas and how does it differ from a real gas?

An ideal gas is a theoretical gas that follows the Ideal Gas Law exactly under all conditions. It is assumed to have no volume and no intermolecular forces, and its particles are considered to be point masses. In reality, most gases deviate from ideal behavior due to factors such as intermolecular forces and the finite size of gas particles. Real gases also have non-zero volumes, which can affect their behavior under high pressures or low temperatures.

5. How is the Ideal Gas Law used in scientific research and applications?

The Ideal Gas Law is used in a wide range of scientific fields, including chemistry, physics, and engineering. It is commonly used to predict the behavior of gases under different conditions, such as changes in temperature, pressure, or volume. It is also used in various industrial applications, such as in the design of gas storage tanks and in the production of various gases. Additionally, the Ideal Gas Law is used in the study of weather and atmospheric conditions, as well as in the development of new materials and technologies.

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