Question

Cylinder Pressure If gas in a cylinder is maintained at a constant temperature \(T,\) the pressure \(P\) is related to the volume \(V\) by a formula of the form $$P=\frac{n R T}{V-n b}-\frac{a n^{2}}{V^{2}}$$ in which \(a, b, n,\) and \(R\) are constants. Find \(d P / d V\) .

Short answer

The derivative of the pressure with respect to volume, holding temperature constant is given by \(\frac{dP}{dV} = \frac{-n^{2}RT}{(V-nb)^{2}} + \frac{2an^{2}}{V^{3}}\)

Step-by-step solution

Identify the Parts of the Equation

From the given equation, the parts can be identified as \(P = f(V) = \frac{n R T}{V-n b}-\frac{a n^{2}}{V^{2}}\).

Component-wise Differentiation

Differentiate the function with respect to \(V\). Using the quotient rule we get the following results: \(\frac{d}{dx}\frac{u}{v} = \frac{u'v - uv'}{v^{2}}\). Applying it to the first term developed from the equation, \(\frac{nRT}{V-nb}\), where \(u = nRT\) and \(v = V - nb\), we get \(\frac{-n^{2}RT}{(V-nb)^{2}}\). Same method applied to the second term derived from the equation, \(-\frac{an^{2}}{V^{2}}\), where \(u = -an^{2}\) and \(v = V^{2}\), we get \(\frac{2an^{2}}{V^{3}}\).

Combine Results and Simplify

By combining the results from step 2, for the full derivative, \(\frac{dP}{dV}\) we get \(\frac{dP}{dV} = \frac{-n^{2}RT}{(V-nb)^{2}} + \frac{2an^{2}}{V^{3}}\)

Key concepts

Quotient Rule Differentiation

In differential calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. It is particularly useful when dealing with complex functions where one variable depends on another in the format of \( \frac{u}{v} \) where \( u \) and \( v \) are both functions of \( x \). To differentiate such a function, the quotient rule formula states: \[ \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^{2}} \]

• \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively with respect to \( x \).
• \( u'v \) denotes the product of the derivative of \( u \) and the function \( v \).
• \( uv' \) denotes the product of the function \( u \) and the derivative of \( v \).
• The denominator \( v^{2} \) is the square of the function \( v \).

Differential of a Function

The differential of a function provides an approximation of how a function's output value changes in response to a change in the input value. When the function is differentiable, the differential, denoted by \( df \) or \( dy \) for a function \( f \) or \( y \) with respect to \( x \) is defined as: \[ df = f'(x)dx \] where \( f'(x) \) is the derivative of the function at point \( x \) and \( dx \) is an infinitesimally small change in the input value \( x \). This concept is fundamental in determining rates of change and is used in a variety of applications, including physics, engineering, and economics to model behavior around a specific point.

Cylinder Pressure and Volume Relationship

In thermodynamics, the relationship between the pressure \( P \) and volume \( V \) of gas in a cylinder can be described by the ideal gas law when temperature \( T \) and the number of moles \( n \) are constant. However, the given equation \[ P=\frac{n R T}{V-n b}-\frac{a n^{2}}{V^{2}} \] accounts for real gas behavior by incorporating constants \( a \) and \( b \) which correct for the volume occupied by the gas molecules and the intermolecular forces, respectively.
The differential \( \frac{dP}{dV} \) reflects the rate of change of pressure as the volume changes. A negative value of \( \frac{dP}{dV} \) signifies that the pressure decreases as the volume increases, which aligns with the general behavior of gases (as per Boyle's Law in ideal conditions). This understanding is essential in calculating work done by or on the gas during expansion or compression.