Which of the following is not an exact differential?

Answer options :

qp

qv

dq/T

dq+dw

answer with explanation

To determine which of the given options is not an exact differential, we first need to understand what an exact differential is. An exact differential is a differential expression that corresponds to the total differential of a function. In simpler terms, if a differential form can be expressed as the derivative of some function, it is considered exact.

Understanding Exact Differentials

For a differential \( df \) to be exact, it must satisfy the condition that the mixed partial derivatives are equal. For instance, if we have a function \( f(x, y) \), the total differential is given by:

df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy

For this to be an exact differential, it must hold that:

\(\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}\)

Analyzing the Options

• qp: This represents the product of two variables, which can be derived from a function, making it an exact differential.
• qv: Similar to qp, this also represents a product of variables and can be associated with a function, thus it is an exact differential.
• dq/T: This expression involves a differential divided by a temperature term. It does not correspond to a simple total differential of a function, as it introduces a division that complicates the relationship between the variables.
• dq + dw: This is a sum of differentials and can represent the total differential of a function of two variables, making it an exact differential.

Identifying the Non-Exact Differential

From our analysis, the expression dq/T stands out as it does not fit the criteria for being an exact differential. The division by T complicates the relationship and does not allow it to be expressed as the total differential of a function. Therefore, it does not satisfy the necessary conditions for exactness.

In summary, the option that is not an exact differential is dq/T. Understanding the nature of differentials and their relationships to functions is crucial in fields like thermodynamics and calculus, where these concepts frequently arise.