Question

Ideal Gas Law The Ideal Gas Law is \(p V=m R T,\) where \(R\) is a constant, \(m\) is a constant mass, and \(p\) and \(V\) are functions of time. Find \(d T / d t,\) the rate at which the temperature changes with respect to time.

Short answer

\( \frac{dT}{dt} = \frac{\frac{dp}{dt}V + p\frac{dV}{dt}}{mR} \)

Step-by-step solution

Write the Ideal Gas Law

The Ideal Gas Law is given by the equation \( pV = mRT \).

Differentiate with respect to time

We need to differentiate the Ideal Gas Law equation with respect to time (t). Using the product rule, the derivative of \( pV \) with respect to \( t \) is \( \frac{dp}{dt}V + p\frac{dV}{dt} \). The derivative of the right side \( mRT \) with respect to \( t \) is \( mR\frac{dT}{dt} \). So, we get the equation \( \frac{dp}{dt}V + p\frac{dV}{dt} = mR\frac{dT}{dt} \).

Solve for \( \frac{dT}{dt} \)

Rearrange the equation from Step 2 to solve for \( \frac{dT}{dt} \). Divide both sides of the equation by \( mR \), which results in \( \frac{dT}{dt} = \frac{\frac{dp}{dt}V + p\frac{dV}{dt}}{mR} \). This gives the rate at which temperature changes with respect to time.

Key concepts

Differentiate Ideal Gas Law

The Ideal Gas Law, represented by the equation \( pV = mRT \), is a fundamental relation in thermodynamics and physics, describing the behavior of ideal gases. To differentiate this law with respect to time, we treat pressure \( p \) and volume \( V \) as variables that can change over time, while mass \( m \) and the ideal gas constant \( R \) remain constant.
When differentiating a product of two functions that both depend on time—like pressure and volume—we must use the product rule from calculus. In the context of the Ideal Gas Law, this differentiation process helps us understand how changing conditions affect the temperature of the gas.
Be sure to remember that while mass and the gas constant are fixed values in this scenario, differentiating an equation with multiple variables requires careful application of calculus rules to obtain accurate results. This step is crucial for solving problems that involve dynamic systems, where variables change over time.

Rate of Temperature Change Calculus

In calculus, finding the rate of change of a quantity with respect to time is a common task, and this problem illustrates how we can determine the rate at which the temperature \( T \) of an ideal gas changes over time. By differentiating the Ideal Gas Law with respect to time and rearranging the equation, we obtain an expression for \( \frac{dT}{dt} \), the rate of temperature change.
Understanding the rate of temperature change is particularly important in various scientific and engineering disciplines. It can provide insights into thermal processes, reaction kinetics, and systems' behavior under varying conditions.
Keep in mind the practical applications, such as monitoring the temperature of gases in real-time in industrial processes, or predicting how atmospheric conditions might change. The calculus-based approach we use here is a powerful tool that can be extended to many similar problems, enabling us to model and predict changes in a wide range of scenarios.

Product Rule Differentiation

The product rule is an essential differentiation technique in calculus used when we want to find the derivative of the product of two functions. The rule states that if you have a product of two functions, say \( u(t) \times v(t) \), the derivative of this product with respect to time \( t \) is given by \( u'(t) \times v(t) + u(t) \times v'(t) \).
It's important to understand that each term in the product has its contribution to the rate of change of the whole. In the Ideal Gas Law differentiation, pressure \( p \) and volume \( V \) are the two functions of time, and applying the product rule allows us to account for how their individual rates of change contribute to the change in temperature.
Whenever faced with differentiating a product of functions, especially in physics or engineering problems, think of the product rule as your go-to tool. It simplifies complex relations and provides a clear path to finding the rates of change of related variables.