Heat capacity is the amount of heat needed to raise the temperature of a substance. There are two types : Cp (at constant pressure ) and Cv (at constant volume). The key difference is that when a substance is heated at constant pressure (Cp), it also does same work by expanding,

While at constant volume (Cv), no expansion happens, so all the heat goes into raising the temperature. That’s why Cp is always greater than Cv. This relationship is especially important

In gases, it helps us understand how energy is transferred in different conditions.

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Define Heat Capacity

The concept of heat capacity is central to thermodynamics. It's all about how a body responds to the absorption or loss of heat. When a body absorbs heat, its temperature rises. When it loses heat, the temperature falls. This change in temperature is directly related to the heat transfer.

The equation q = n C ∆T is used to calculate the heat (q) required to change the temperature of one mole of any substance by a certain amount (∆T). The constant C in this equation is known as the molar heat capacity of the substance. It represents the amount of heat energy needed to change the temperature of 1 mole of a substance by 1 degree. This value depends on factors such as the substance's composition, size, and nature.

In this article, we'll focus on two specific types of molar heat capacity - CP and CV - and explore their relationship.

• The molar heat capacity C at constant pressure is denoted by CP.
• The molar heat capacity C at constant volume is denoted by CV.

The Relationship between CP and CV for an Ideal Gas

Let's start with the equation q = n C ∆T. From this, we can derive:

At constant pressure P, the equation becomes qP = n CP ∆T. This is equivalent to the change in enthalpy (∆H).

Similarly, at constant volume V, the equation becomes qV = n CV ∆T. This is equivalent to the change in internal energy (∆U).

For one mole of an ideal gas (n=1), we know that ∆H = ∆U + ∆(pV) = ∆U + ∆(RT) = ∆U + R ∆T.

So, we can say that ∆H = ∆U + R ∆T.

Substituting the above values for ∆H and ∆U into the equation, we get:

CP ∆T = CV ∆T + R ∆T
CP = CV + R
CP – CV = R

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