Stat Mech- How does heat vary with temperature over an arbitrary pathway

T2):(P1)(V1)/T1 = (P2)(2V1)/T2Simplifying this equation, we get:T2 = 2T1This tells us that the final temperature (T2) is twice the initial temperature (T1). In other words, the temperature must double in order for the volume of the gas to double.In summary, we have 1 mole of an ideal gas with a heat capacity at constant volume of 1.5R. The heat flow rate along the pathway is 2R, and the volume of the gas doubles. By using the ideal gas law and considering the reversible process, we can conclude that the final temperature must double in order for the
  • #1
mandir08
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Homework Statement



You are told that you have 1 mole of an ideal gas with heat capacity at constant volume being 1.5R and you send it over an arbitrary path where dq/dt|pathway= 2R. In the end, the volume of the gas doubles, so figure out by what factor the temperature must change. Assume that the process is reversible


The Attempt at a Solution




du=dq +dw= dq -p*dv
dq=du + p*dv
dq/dt|path =du/dt |path + (d/dt* (P)* dv + P dv/dt|path)

Saying that P= -dF/dv|t,n cause it to become

dq/dt|path =du/dt |path + (0+ P dv/dt|path)

dq/dt|path=2R

u=Cv*T=NRC *T (I am not sure I can apply equation of state u=Cv*T
du/dt=Cv=1.5R =NRC

2R =du/dt |path + (0+ P dv/dt|path)
2R =1.5R + (0+ P dv/dt|path)=RC + (0+ P dv/dt)
.5R=P* dv/dt|path
P/.5R=dt/dv|path

V*P/.5R=T such that T2=V2*P2/.5R=2V1*P2/.5R

However because I am sure that I did this wrong because I can get the same result knowing that PV=NRT such that T2=2*(P2/P1) T1

The hint at the end about the process being reversible makes it that Cv=(du/dt)|v=T(ds/dt)|v

I am kind of lost at this point and not really sure how to proceed. Anyone want to help point me in the right direction?
 
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  • #2


it's important to approach problems like this with a systematic and logical approach. Let's break down the information given and see what we can deduce from it.

1. We have 1 mole of an ideal gas.
2. The heat capacity at constant volume (Cv) is 1.5R.
3. The heat flow rate along the pathway (dq/dt|pathway) is 2R.
4. The volume of the gas doubles.
5. The process is reversible.

First, let's recall the definition of heat capacity at constant volume (Cv). It is the amount of heat required to raise the temperature of 1 mole of a substance by 1 Kelvin at constant volume. So in this case, we know that the heat flow rate (dq/dt) is equal to 2R. This means that for every second, 2R amount of heat is flowing into our system.

Next, let's consider the fact that the gas is undergoing a reversible process. This means that the gas is always in equilibrium with its surroundings, and the system is always in a state of thermodynamic equilibrium. This is important because it allows us to use the equation of state for an ideal gas, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature.

Now, let's think about what happens to the gas as its volume doubles. We know that the number of moles (n) and the gas constant (R) are constant, so the only variable that can change is the temperature (T). Using the ideal gas law, we can set up the following equation:

(P1)(V1) = (P2)(V2)

where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume. We also know that the final volume (V2) is twice the initial volume (V1), so we can rewrite the equation as:

(P1)(V1) = (P2)(2V1)

Next, we can rearrange the equation to solve for the final pressure (P2):

P2 = (P1)(V1)/(2V1) = P1/2

Since we know that the pressure and number of moles are constant, we can also use the ideal gas law to solve for the final temperature (
 

1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to predict the behavior of large groups of particles, such as atoms or molecules. It is based on the laws of thermodynamics and describes how the properties of a system change at a microscopic level.

2. How does statistical mechanics relate to temperature?

Statistical mechanics helps us understand how the temperature of a system affects the behavior of its particles. It describes the relationship between temperature and the average kinetic energy of the particles in a system.

3. What is the connection between heat and temperature in statistical mechanics?

In statistical mechanics, heat is defined as the transfer of energy from one system to another due to a temperature difference. The amount of heat transferred is directly proportional to the temperature difference between the two systems.

4. How does heat vary with temperature over an arbitrary pathway?

The relationship between heat and temperature over an arbitrary pathway is described by the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. Therefore, the amount of heat transferred will vary depending on the specific pathway taken by the system.

5. How does statistical mechanics explain the behavior of gases?

Statistical mechanics provides a theoretical framework for understanding the behavior of gases, such as how they expand when heated and contract when cooled. It explains these observations by considering the movements and interactions of individual gas particles at a microscopic level.

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