Heat integral and molar heat capacity?

In summary, the conversation discusses the equation dQ = nCvdT and the assumption of a closed system, where n is constant. The speaker questions the validity of this assumption and suggests that the temperature should also be considered in the equation. The concept of heat capacity per mole is also mentioned. The conversation concludes that the assumption of a closed system is a typical requirement when discussing heat capacity, and this is not a result of the second law.
  • #1
Inertigratus
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dQ = nCvdT if volume is constant.
However, n = pV/RT.
What I don't understand is, why are we thinking n as constant when doing the integral?
I had two problems that involved this on a test I had today. At first I kept it constant and then changed n. But then I thought, wait... isn't there a T in n? then that T should be in the integral.
I understand the point, heat capacity per mole. But mathematically, the T that is in the equation for n should matter, right?
dS = dQ / T, if we substitute dQ in that equation we should get 1 / T2 in the integral also.
I know I'm wrong however, so if someone could tell me what's wrong?
 
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  • #2
The equation dQ = nCvdT also assumes a closed system (i.e., constant n). Otherwise you could effect a temperature change by simply removing gas molecules at constant volume without heating or cooling the system, and this would violate the equation.
 
  • #3
That's true... higher temperature with lower amount of moles doesn't sound right. When you say that it assumes a closed system, is that a result of the second law?
Or do we always speak of closed systems when talking about heat capacity?
 
  • #4
It's not a result of the second law. It's a typical assumption of the definition of heat capacity; that is, we mean

[tex]C_{V,N}=T\left(\frac{\partial S}{\partial T}\right)_{V,N}[/tex]

but we generally just write [itex]C_V[/itex].

(In some esoteric circumstances, we want to work with systems at constant chemical potential rather than constant matter, but that's an advanced topic.)
 
  • #5


I can understand your confusion about the heat integral and molar heat capacity. Let me try to explain it in a simpler way.

The heat integral (dQ) is the amount of heat transferred to a system during a process. It is directly proportional to the change in temperature (dT) and the number of moles (n) of the substance being heated. This can be mathematically represented as dQ = nCvdT, where Cv is the molar heat capacity at constant volume.

Now, in the equation you mentioned, n = pV/RT, the temperature (T) is already included in the ideal gas law (pV = nRT). Therefore, when we substitute this value of n in the heat integral equation, we don't need to include T separately in the integral. This is because the change in temperature is already accounted for in the ideal gas law.

To understand this better, let's take an example. Suppose we have a balloon filled with 1 mole of an ideal gas at a constant volume. If we heat the balloon, the temperature will increase and the balloon will expand. In this case, the heat transferred (dQ) will be equal to nCvdT, where n is constant (1 mole) and Cv is the molar heat capacity at constant volume. So, even though the volume is changing, n remains constant and the change in temperature is accounted for in the integral.

In the second equation you mentioned, dS = dQ / T, the temperature (T) is already included in the denominator. So, when we substitute the value of dQ from the heat integral equation, we don't need to include T in the integral again.

I hope this explanation helps to clear your confusion. Just remember, in thermodynamics, the ideal gas law takes into account the changes in temperature and other variables, so we don't need to include them separately in the equations.
 

1. What is the heat integral?

The heat integral is a measure of the total heat energy required to raise the temperature of a substance from one state to another. It takes into account both the initial and final temperatures, as well as the specific heat capacity of the substance.

2. How is the heat integral calculated?

The heat integral is calculated by multiplying the specific heat capacity of the substance by the change in temperature (ΔT) and the mass of the substance. This can be represented by the equation Q = mcΔT, where Q is the heat integral, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

3. What is molar heat capacity?

Molar heat capacity is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius. It is represented by the symbol Cm and is measured in units of joules per mole per degree Celsius (J/mol·°C).

4. How is molar heat capacity different from specific heat capacity?

While both molar heat capacity and specific heat capacity measure the amount of heat energy required to raise the temperature of a substance, they differ in their units. Specific heat capacity is measured per unit mass, while molar heat capacity is measured per mole. Additionally, molar heat capacity takes into account the molar mass of the substance, while specific heat capacity does not.

5. How does heat integral and molar heat capacity relate to each other?

The heat integral and molar heat capacity are related in that the molar heat capacity can be used to calculate the heat integral for a particular substance. By multiplying the molar heat capacity by the change in temperature and the number of moles of the substance, the heat integral can be determined. This relationship is represented by the equation Q = nCmΔT, where Q is the heat integral, n is the number of moles, Cm is the molar heat capacity, and ΔT is the change in temperature.

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