Work Done in a polytropic process

[juggalomike]

Homework Statement

A gas is trapped in a cylinder-piston arrangement. The initial pressure and volume are 13,789.5 Pa and 0.02832 m^3. Determine the work(kj) assuming that the volume is increased to 0.08496 m^3 in a polytropic process with n=1.

Homework Equations

m_1-2=∫PdV from 1 to 2

The Attempt at a Solution

I believe the equation for n=1 i would use is P1*V1*ln(V2/V1), however i have to show how i would get that from the intial integral, and i am completely lost as to how i can get there.

Also if this is not the correct forum for this post i am sorry, please let me know and i will
re-post it in the correct location.

 

[NascentOxygen]

The ln term comes about because the integral of 1/x is ln(x).
Search wikibooks for polytropic process.

 

[Andrew Mason]

Homework Statement

A gas is trapped in a cylinder-piston arrangement. The initial pressure and volume are 13,789.5 Pa and 0.02832 m^3. Determine the work(kj) assuming that the volume is increased to 0.08496 m^3 in a polytropic process with n=1.

Homework Equations

m_1-2=∫PdV from 1 to 2

The Attempt at a Solution

I believe the equation for n=1 i would use is P1*V1*ln(V2/V1), however i have to show how i would get that from the intial integral, and i am completely lost as to how i can get there.

Also if this is not the correct forum for this post i am sorry, please let me know and i will
re-post it in the correct location.

Express P as a function of V and integrate. Since n=1 in $PV^n = K$, this is rather simple.

AM

 

[juggalomike]

Express P as a function of V and integrate. Since n=1 in $PV^n = K$, this is rather simple.

AM

Thanks a lot, knew it wasn't a hard question was just stuck on the sub part.

 

[rwisz]

I would like to clarify that the term "work" in this context refers to thermodynamic work, which is the energy transferred to or from a system due to a change in its volume. The equation for work done in a polytropic process is W = ∫PdV, where P is the pressure and V is the volume. In this case, the value of n=1 indicates that the process is isothermal, meaning that the temperature remains constant throughout the process.

To solve for the work done, we can use the equation W = P1*V1*ln(V2/V1), where P1 is the initial pressure and V1 is the initial volume, and V2 is the final volume in the process. Plugging in the given values, we get W = 13,789.5 Pa * 0.02832 m^3 * ln(0.08496 m^3 / 0.02832 m^3) = 1.47 kJ. This represents the amount of energy transferred to the gas during the process.

To show how this equation is derived from the initial integral, we can start with the general equation for work done in a polytropic process, W = ∫PdV. Since n=1, we can rewrite this as W = ∫P1(V/V1)dV. We can then rearrange this as W = P1*V1*∫(V/V1)dV. By integrating, we get W = P1*V1*(V2-V1)/V1 = P1*V1*(V2/V1 - 1). Since V2/V1 is equal to 3 in this case, the equation simplifies to W = P1*V1*ln(3). Plugging in the given values for P1 and V1, we get the same result of 1.47 kJ.

In conclusion, the work done in this polytropic process is 1.47 kJ. It is important to note that this value only represents the energy transferred to the gas, and does not take into account any other forms of energy, such as heat or internal energy. As a scientist, it is important to consider all factors and variables in a system to fully understand and analyze a process.