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## Homework Statement

Two copper cylinders, immersed in a water tank at

**30.3 °C**, contain helium and nitrogen, respectively. The helium-filled cylinder has a volume twice as large as the nitrogen-filled cylinder.

**a)**Calculate the average translational kinetic energy of a helium molecule and the average translational kinetic energy of a nitrogen molecule.

**b)**Determine the molar specific heat at constant volume (C

_{V}) and at constant pressure (C

_{p}) for the two gases.

**c)**Find

**γ**for the two gases.

**Given/Known information****C**

_{p}= C_{v}+R**C**monatomic gas

_{V}= [itex]\frac{3}{2}[/itex]R = 12.5 J/(mol*K)**C**monatomic gas

_{p}= [itex]\frac{5}{2}[/itex]R = 20.8 J/(mol*K)**C**Nitrogen gas

_{V}= 20.7 J/(mol*K)**C**Nitrogen gas

_{p}= 29.1 J/(mol*K)**T = 30.3 °C = 303.3 K**

## Homework Equations

*K*_{ave}= [itex]\frac{3}{2}[/itex]*k*_{B}T**γ = C**

_{p}/C_{V}## The Attempt at a Solution

__Part a)__Helium is a monatomic molecule.

The average energy per degree of freedom is given by [itex]\frac{1}{2}[/itex]

*k*

_{B}for each gas molecule. I was asked for the average

**translational**kinetic energy, since a monatomic molecule has 3 translational degrees of freedom, and Nitrogen has 3 translational degrees of freedom and 2

**rotational**degrees, then the average kinetic energy is equal to,

__Helium__**=[itex]\frac{3}{2}[/itex]**

*K*_{ave}

*k*_{B}T**=[itex]\frac{3}{2}[/itex]([itex]1.38\ \times\ 10^{-23}\ J\ K^{-1}[/itex])**

*K*_{ave}**(303.3 K)**

**=[itex]6.278\ \times\ 10^{-21}\ J[/itex] ←**

*K*_{ave}

*INCORRECT*

__Nitrogen__**=[itex]\frac{3}{2}[/itex]([itex]1.38\ \times\ 10^{-23}\ J\ K^{-1}[/itex])**

*K*_{ave}**(303.3 K)**

**=[itex]6.278\ \times\ 10^{-21}\ J[/itex] ←**

*K*_{ave}

*INCORRECT*I don't know if the two gases being inside of copper cylinders affects the calculations somehow since there is no temperature change, no reaction going on, or anything that will change the equation. I know that for a polyatomic molecule you need to consider the additional degrees of freedom available, but for both

**He**and

**N**, there are 3

_{2}**translational**degrees of freedom, giving us the [itex]\frac{3}{2}[/itex]. I don't know why that one is wrong.

__Part b)__Values were obtained from a table in the book on Molar Specific Heats at constant volume and constant pressure. Also on this table was a column for the values of

**γ=C**for some gases.

_{p}/C_{V}**C**

C

C

C

_{V, He}= 12.5 J/(mol*K) ←CORRECTC

_{p, He}= 20.8 J/(mol*K) ←CORRECTC

_{V, N2}= 20.7 J/(mol*K) ←CORRECTC

_{p, N2}= 29.1 J/(mol*K) ←CORRECT

__Part c)__**γ**is stated to be the ratio

**C**

_{p}/C_{V}**γHe = C**

γHe = (20.8 J/(mol*K))/(12.5 J/(mol*K))

γHe = 1.664 ≈ 1.67←

_{p, He}/C_{V, He}γHe = (20.8 J/(mol*K))/(12.5 J/(mol*K))

γHe = 1.664 ≈ 1.67

*INCORRECT***γN**

γN

γN

_{2}= C_{p, N2}/C_{V, N2}γN

_{2}= (29.1 J/(mol*K))/(20.7 J/(mol*K))γN

_{2}= 1.4058 ≈ 1.4 ←CORRECTI really don't know what other value

**γHe**could be if we are told

**γHe=C**. Using values that I verified are correct for part b, taking

_{p}/C_{V}**C**for

_{p}/C_{V}**He**should give me the correct answer, which the table in the book also verifies that it is a value of

**1.67**. Even more confusing is how doing that calculation for

**N**gives me the correct answer for

_{2}**γN**.

_{2}All help is greatly appreciated.

Thanks in advance

Please let me know if something is confusing or if something looks like a typo, this took me a little while to proofread and edit and I went back numerous times to make it easier to read so I may have missed something.