Find temperature of a U-shaped Tube

[hraghav]

Homework Statement

A simple thermometer is formed by taking a large spherical container of volume 1.81m^3 and filling it with an ideal gas. Then, a U-shaped tube is attached that is very narrow so that very little fluid/gas can fit inside the tube. The tube has a small quantity of a fluid of density 2560kg/m^3 placed in it. In equilibrium at room temperature (that is at 24.3∘C) the height of the fluid on both sides of the U-shaped tube are level. This is illustrated in the image below. What is the temperature of the gas in the spherical container when the left side of the fluid is 8.5cm higher than the right side of the fluid in the U-shaped tube? Atmospheric pressure is 101000Pa.

Relevant Equations

PV = nRT
P = density*g*h

So first I used the ideal gas law formula PV = nRT to find n using P = 101000Pa, V = 1.81m^3, R = 8.314 J/mol*K^-1, T = 24.3+273.15 = 297.45K. I got n = 73.922 mol.
Then I used P = density*g*h and found P = 2132.48 where density = 2560kg/m^3, g = 9.8 and h = 0.085m.
Then I found total pressure by adding 101000 + 2132.48 = 103132.48
Finally I used the ideal gas law formula PV = nRT again to find T where P = 103132.48, V = 1.81m^3, R = 8.314 J/mol*K^-1 and n = 73.922 mol. I got T = 303.7318K and then converted that to degree C and got 30.58.

But this is not the correct answer and I have attempted this several times but I still can't find the error. Could someone please tell me where am I going wrong?

Thanks a lot for your help! Much appreciated :)

 

[TSny]

Then I found total pressure by adding 101000 + 2132.48 = 103132.48

Is the final pressure in the spherical container greater or less than the initial pressure? Why?

 

[hraghav]

Im sorry I am not sure what you mean

Is the final pressure in the spherical container greater or less than the initial pressure? Why?

 

[TSny]

Im sorry I am not sure what you mean

Based on the information given about the final level of the liquid on each side of the U-tube, did the pressure in the spherical container increase or decrease?

 

[hraghav]

It increases as the left side of the fluid is 8.5cm higher than the right side

Based on the information given about the final level of the liquid on each side of the U-tube, did the pressure in the spherical container increase or decreases?

 

[TSny]

It increases as the left side of the fluid is 8.5cm higher than the right side

If a straw is placed in a glass of water, the water level is the same inside and outside the straw. If I start to drink through the straw, the water inside the straw rises. Did I Increase the pressure in my lungs or decrease the pressure in my lungs?

 

[hraghav]

If a straw is placed in a glass of water, the water level is the same inside and outside the straw. If I start to drink through the straw, the water inside the straw rises. Did I Increase the pressure in my lungs or decrease the pressure in my lungs?

The pressure inside the straw decreases as we drink as it allows the water to rise up the straw.

 

[TSny]

The pressure inside the straw decreases as we drink as it allows the water to rise up the straw.

Yes. Can you relate this to the homework question?

 

[hraghav]

Yes. Can you relate this to the homework question?

Yes it makes sense now I had to subtract not add thanks!

 

[TSny]

The value of the volume of the container is not needed. $n$ and $R$ are constant and $V$ is taken to be approximately constant.

The ideal gas law gives $$\frac P T = \frac{nR}{V} = \rm{constant} $$ So, if $P_i$, $T_i$ are the initial pressure and temperature and $P_f$, $T_f$ the final pressure and temperature, $$\frac{P_f}{T_f} = \frac {P_i}{T_i}.$$ The final temperature $T_f$ can be found from this equation.