Thermal Equilibrium With Insulated Liquid And Gas Containers

In summary: You can solve for the final temperature, and then use the ideal gas law to find the final pressure.In summary, the problem involves a beaker of water at 20∘C and a container of monatomic gas with 0.40mol and a pressure of 10atm, both insulated. The goal is to determine the gas pressure after a long time has elapsed. Using the heat capacities of water and the gas, the final temperature can be solved for, and then the ideal gas law can be used to find the final pressure.
  • #1
mchahal22
2
0

Homework Statement


A beaker with a metal bottom is filled with 20g of water at 20∘C. It is brought into good thermal contact with a 4000 cm^3 container holding 0.40mol of a monatomic gas at 10atm pressure. Both containers are well insulated from their surroundings.

What is the gas pressure after a long time has elapsed? You can assume that the containers themselves are nearly massless and do not affect the outcome.

knight_Figure_17_48.jpg


Homework Equations


Q=mc(delta T)
P=kA(delta T/distance between objects)
pV=nRT


The Attempt at a Solution


I used the ideal gas law to find the initial temp of the gas to be 1219.013 K or 945.863 degrees Celsius. From there, I do not know how to equate the substances in the two containers to determine an equilibrium point and find the final pressure or if this is even the right approach. I would greatly appreciate any help.
 
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  • #2
mchahal22 said:

Homework Statement


A beaker with a metal bottom is filled with 20g of water at 20∘C. It is brought into good thermal contact with a 4000 cm^3 container holding 0.40mol of a monatomic gas at 10atm pressure. Both containers are well insulated from their surroundings.

What is the gas pressure after a long time has elapsed? You can assume that the containers themselves are nearly massless and do not affect the outcome.

knight_Figure_17_48.jpg


Homework Equations


Q=mc(delta T)
P=kA(delta T/distance between objects)
pV=nRT


The Attempt at a Solution


I used the ideal gas law to find the initial temp of the gas to be 1219.013 K or 945.863 degrees Celsius. From there, I do not know how to equate the substances in the two containers to determine an equilibrium point and find the final pressure or if this is even the right approach. I would greatly appreciate any help.
The heat capacity of the water is 1 cal/(gm C) and the molar heat capacity of the gas is 3R/2 = 3 cal/(mole C). The amount of heat gained by the water is equal to the amount of heat lost by the gas. Their final temperatures are the same.

Chet
 
  • #3
Chestermiller said:
The heat capacity of the water is 1 cal/(gm C) and the molar heat capacity of the gas is 3R/2 = 3 cal/(mole C). The amount of heat gained by the water is equal to the amount of heat lost by the gas. Their final temperatures are the same.

Chet

So I set the two amounts of heat equal to each other:

mc deltaT (water) = mc delta T (gas)

.02 kg x 4190 J/kgC x (T-20) = - (T-945.863) x m x c

I'm not really getting what m and c would be for the right side of the equation.

m x c = .4 mol x (3x4190) J/moleC ? Is that the correct conversion?
 
  • #4
mchahal22 said:
So I set the two amounts of heat equal to each other:

mc deltaT (water) = mc delta T (gas)

.02 kg x 4190 J/kgC x (T-20) = - (T-945.863) x m x c

I'm not really getting what m and c would be for the right side of the equation.

m x c = .4 mol x (3x4190) J/moleC ? Is that the correct conversion?
Yes. But it would have been easier if you stuck to calories.

20 gm x 1 cal/gmC x (T-20) = - (T-945.863) x .4 mol x 3 cal/moleC
 
  • #5


I would first clarify the problem by asking some questions. Are the containers initially at the same temperature? Are the containers in thermal contact with each other or just the beaker with the metal bottom in contact with the gas container? Is the gas container insulated as well or just the beaker? These details will affect the approach to finding the final pressure.

Assuming that the containers are initially at the same temperature and are in thermal contact with each other, we can use the concept of thermal equilibrium to determine the final pressure. Thermal equilibrium is when two objects in contact with each other reach the same temperature. In this case, the beaker and the gas container will reach the same final temperature.

We can use the equation Q = mcΔT to calculate the amount of heat transferred between the two containers. Since the containers are well insulated, we can assume that there is no heat lost to the surroundings. Therefore, the heat lost by the gas container must be equal to the heat gained by the beaker.

We can also use the equation pV = nRT to find the initial and final pressures of the gas. Since the temperature is constant, we can set the initial and final pressures equal to each other and solve for the final pressure.

Using these equations and assumptions, we can determine the final pressure of the gas after a long time has elapsed. However, if the containers are not initially at the same temperature or if they are not in thermal contact with each other, a different approach would be needed.
 

1. How do insulated containers maintain thermal equilibrium?

Insulated containers maintain thermal equilibrium by preventing the transfer of heat between the container and its surroundings. This is achieved through the use of insulating materials, such as foam or vacuum-sealed layers, which reduce heat transfer through conduction, convection, and radiation.

2. What is the purpose of thermal equilibrium in liquid and gas containers?

The purpose of thermal equilibrium in liquid and gas containers is to maintain a constant temperature within the container, regardless of the temperature outside. This is important for storing and transporting temperature-sensitive materials, as well as for conducting accurate experiments and measurements.

3. How does thermal equilibrium affect the behavior of gases and liquids?

In thermal equilibrium, the temperature of the gas or liquid in the container remains constant, resulting in a balance between the kinetic energy of the particles and the surrounding environment. This can affect the behavior of the substances, such as the rate of evaporation or condensation, and the pressure and volume of the gas.

4. What factors can disrupt thermal equilibrium in insulated containers?

Thermal equilibrium can be disrupted by factors such as changes in external temperature, exposure to sunlight or other heat sources, and physical damage to the insulation. Additionally, the material and design of the container can also affect the ability to maintain thermal equilibrium.

5. How can thermal equilibrium be measured and monitored in insulated containers?

Thermal equilibrium can be measured and monitored using thermometers or temperature sensors placed inside the container. Additionally, changes in pressure or volume of the gas can also indicate a disruption in thermal equilibrium. Regular inspections and maintenance of the insulation can also help ensure the container is maintaining thermal equilibrium.

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