Polytropic Process: Solving for p1 with p2, T1, T2 & n

[Tux9876]

Hi,

I have a question relating to a polytropic process. The question consists of two parts.

It says that the pressure (p) and the volume (v) of a gas undergoing a polytropic process are related by the equation:

p1V1^n = p2V2^n (where P1, P2 etc are individual variables as i don't know how to put sub text into the question)
the question then says...where n is the polytropic index.

IF

p1V1/T1 = p2V2/T2

determine an expression for p!, in terms of p2,T1,T2 and n

I have had a look on the net for help in answering the question but not luck so far.

 

[FredGarvin]

I am assuming that you want to find P1 and not P factorial. This should be basic algebra (I hope). It looks like it could be a bit of a trick question because usually you just deal with P and V in a polytropic compression. To calculate T you need to know about the substance.

If $$P_1 V_1^n = P_2 V_2^n$$ Then

$$\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}$$

$$\frac{P_1 V_1^n}{T_1}=\frac{P_2 V_2^n}{T_2}$$

$$P_1 = \frac{P_2 T_1}{T_2} \left[ \frac{V_2}{V_1} \right] ^n$$

 

[piercas]

Hi there,

I can provide some insight into your question about the polytropic process. The polytropic process is a thermodynamic process in which the pressure and volume of a system are related by the equation p1V1^n = p2V2^n, where n is the polytropic index.

In order to solve for p1 in terms of p2, T1, T2, and n, we can use the ideal gas law, which states that pV = nRT, where p is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.

Using this equation, we can rewrite the given equation as:

p1V1/T1 = p2V2/T2

p1 = (p2V2/T2)*(T1/V1)

Next, we can substitute in the ideal gas law for both p1 and p2:

p1 = (n1RT1/V1)*(T1/V1)

p2 = (n2RT2/V2)*(T2/V2)

Where n1 and n2 are the number of moles of gas at the initial and final states, respectively.

Now, we can combine these equations to solve for p1:

p1 = (n2RT2/V2)*(T2/V2)*(T1/V1)*(V1/n1RT1)

p1 = p2*(T2/T1)*(V1/V2)^(n-1)

Therefore, the expression for p1 in terms of p2, T1, T2, and n is:

p1 = p2*(T2/T1)*(V1/V2)^(n-1)

I hope this helps to answer your question. If you have any further questions, please don't hesitate to ask. Best of luck!