Molar Heat Capacities and Specific Heats for Ideal Gases

In summary, an ideal gas is heated at a constant volume and pressure. The infinitesimal change in internal energy is given by dU = nCvdT. Cp is given by pV = nRT. The specific heat ratio is \gamma.
  • #1
Nachore
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Homework Statement



a. Consider an ideal gas being heated at constant volume, and let Cv be the gas's molar heat capacity at constant volume. If the gas's infinitesimal change in temperature is dT, find the infinitesimal change in internal energy dU of n moles of gas.
Express the infinitesimal change in internal energy in terms of given quantities.

b. Now suppose the ideal gas is being heated while held at constant pressure p. The infinitesimal change in the gas's volume is dV, while its change in temperature is dT. Find the gas's molar heat capacity at constant pressure, Cp.
Express in terms of some or all of the quantities Cv, p, dV, n, and dT.

c. Suppose there are n moles of the ideal gas. Simplify your equation for Cp using the ideal gas equation of state: pV = nRT.
Express Cp in terms of some or all of the quantities Cv, n, and the gas constant R.

d. The ratio of the specific heats Cp/Cv is usually denoted by the Greek letter [tex]\gamma[/tex]. For an ideal gas, find [tex]\gamma[/tex].
Give your answer in terms of some or all of the quantities n, R, and Cv.


Homework Equations


I don't know.


The Attempt at a Solution


For part a, I did dU = nCvdT, but I don't know if it's right.
I'm having trouble approaching rest of the parts. Help please?
 
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  • #2
Part (a) looks fine. When you increase the internal energy of an ideal gas, its temperature increases, and the constant of proportionality is the heat capacity.

At constant volume, all the energy you added went to increase the internal energy of the gas. But at constant pressure, the gas is allowed to expand, which means it's going to do some [itex]p\,dV[/itex] work on the environment. Now the energy you put in will be divided between this work and increasing the internal energy ([itex]dU[/itex]). You'll have both [itex]dU[/itex] and [itex]p\,dV[/itex] where you used to have just [itex]dU[/itex]. Does this help?
 
  • #3
Umm kind of. But I'm having the most trouble from part b through part d.
 
  • #4
My second paragraph is about part (b). Where are you stuck there?
 
  • #5
Thanks for part b. But I don't get part c, d.
 
  • #6
OK, have you applied the ideal gas law to part (b)?
 
  • #7
yes, i got it. thanks
 
  • #8
how do you do part d? does anyone know?
 
  • #9
Help?
 
  • #10
What do you have so far?
 
  • #11
I know [tex]\gamma[/tex] = Cp/Cv
and
Cv = R/([tex]\gamma[/tex] - 1)

How do I find n?
 
  • #12
[itex]\gamma[/itex] is independent of n.
 

What is molar heat capacity?

Molar heat capacity, also known as specific heat, is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius.

What is the difference between molar heat capacity and specific heat capacity?

Molar heat capacity is the amount of heat energy required to raise the temperature of one mole of a substance, while specific heat capacity is the amount of heat energy required to raise the temperature of one gram of a substance. Molar heat capacity is a more useful measurement for scientists, as it allows for comparisons between different substances regardless of their mass.

How is molar heat capacity measured?

Molar heat capacity is typically measured using a calorimeter, which is a device that can measure the heat absorbed or released by a substance during a chemical or physical change. The change in temperature of the substance is then used to calculate its molar heat capacity.

What factors can affect the molar heat capacity of a substance?

The molar heat capacity of a substance can be affected by its chemical composition, temperature, and pressure. It can also vary depending on the phase of the substance (solid, liquid, or gas) and whether the process is reversible or irreversible.

Why is molar heat capacity an important concept in thermodynamics?

Molar heat capacity is an important concept in thermodynamics because it helps us understand how much energy is required to change the temperature of a substance. It is also used in calculations for determining the enthalpy, entropy, and Gibbs free energy of a system, which are key factors in understanding chemical and physical processes.

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