Finding the number of moles of an ideal gas in a capillary

In summary: You have an expression for T as a function of distance. You know how n varies with T and V.It is possible that your second method can be made to work but I cannot see how.In summary, the conversation discusses an equation for the temperature across a capillary and the number of moles in an ideal gas under constant pressure. Two methods for finding the number of moles are examined, but the first method results in a negative volume change and the second method is not applicable in this context.
  • #1
Potatochip911
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Homework Statement


The temperature across the capillary with constant cross-sectional area and length L is given by ##T=T_0e^{-kx}##. Assuming an ideal gas and constant pressure show the number of moles to be: $$n=\frac{PV(e^{kL} - 1)}{RkLT_0}$$

Homework Equations


##PV=nRT##

The Attempt at a Solution



The equation of state can be expressed as ##g(P,V,T) = 0## but since pressure is given to be constant we have ##g(V,T) = 0## therefore we can express the volume as ##V=V(T)## from which we can get the differential for V as $$dV = (\frac{\partial V}{\partial T})dT = \frac{nR}{P}dT\\ V =\frac{nR}{P} \int_{T_i}^{T_f}dT = \frac{nR}{P}(T_f-T_i)$$

Using ##T=T_0e^{-kx}## it is evident that ##T_f = T_0e^{-kL}## and ##T_i = T_0## therefore $$V = \frac{nRT_0}{P}(e^{-kL}-1)\Longrightarrow n=\frac{PV}{nRT_0(e^{-kL}-1)}$$

Which clearly isn't the correct answer. I'm curious as to what the mistake is following this reasoning.

Another method I've attempted is for an ideal gas ##n=n(P,V,T)## but pressure is constant therefore ##n=n(V,T)## and we obtain the differential $$dn = \left(\frac{\partial n}{\partial V}\right)_{P,T} dV + \left(\frac{\partial n}{\partial T}\right)_{P,V} dT$$

Now I know from the the answer that ##\left(\frac{\partial n}{\partial T}\right)_{P,V}## must equal 0 since the answer is obtained from integrating ##dn = \left(\frac{\partial n}{\partial V}\right)_{P,T} dV ## but I can't seem to justify why ##\left(\frac{\partial n}{\partial T}\right)_{P,V}## should be equal to zero without setting both of them equal to zero.
 
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  • #2
Potatochip911 said:
I'm curious as to what the mistake is following this reasoning.
You need to understand what your differential equation says. You have taken a constant n and expressed how the volume changes if you change the temperature. Since the change in temperature was negative, you got a negative volume change.
Consider segments length dx and the number of moles they contain.
 
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  • #3
haruspex said:
You need to understand what your differential equation says. You have taken a constant n and expressed how the volume changes if you change the temperature. Since the change in temperature was negative, you got a negative volume change.
Consider segments length dx and the number of moles they contain.
Edit: Ok so that's the logic behind taking ##dn/dV## and then multiplying the ##dV## over and integrating, but shouldn't this also come from my second method in the main post?
 
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  • #4
Potatochip911 said:
Edit: Ok so that's the logic behind taking ##dn/dV## and then multiplying the ##dV## over and integrating, but shouldn't this also come from my second method in the main post?
Your second method considers how n varies as T and V change independently. I cannot think what that means in the context of the question.
As I posted, your independent variable should be distance along the capillary.
 
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Related to Finding the number of moles of an ideal gas in a capillary

What is an ideal gas?

An ideal gas is a hypothetical gas that follows the ideal gas law, which states that the pressure, volume, and temperature of the gas are all directly proportional.

How do you find the number of moles of an ideal gas in a capillary?

The number of moles of an ideal gas in a capillary can be found by using the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

What is the ideal gas constant?

The ideal gas constant, denoted by the symbol R, is a constant that relates the properties of an ideal gas. Its value is 0.08206 L·atm/mol·K at standard temperature and pressure.

What is a capillary?

A capillary is a thin tube with a small diameter, typically used in laboratory experiments to contain small amounts of liquids or gases.

How is the ideal gas law used in real-life situations?

The ideal gas law is used in many real-life situations, such as calculating the volume of a gas in a sealed container, determining the pressure inside a gas cylinder, and predicting the behavior of gases in chemical reactions. It is also used in industries such as aerospace and pharmaceuticals for designing and testing equipment.

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