Question

Charles' law states that for a fixed mass of gas under constant pressure, the volume \(V\) and temperature \(T\) of the gas (in kelvins) satisfy the equation \(V=k T\), where \(k\) is constant. Find an equation relating \(d V / d t\) to \(d T / d t\)

Short answer

Answer: The relationship between the rate of change of volume with respect to time, \(\frac{dV}{dt}\), and the rate of change of temperature with respect to time, \(\frac{dT}{dt}\), according to Charles' Law, can be expressed as \(\frac{dV}{dt} = k\frac{dT}{dt}\), where \(k\) is a constant.

Step-by-step solution

Write down the given equation

The given equation is Charles' Law, \(V = kT\), where \(V\) is volume, \(T\) is temperature in kelvins, and \(k\) is a constant.

Differentiate both sides of the equation with respect to time

In order to find an equation relating \(\frac{dV}{dt}\) to \(\frac{dT}{dt}\), we need to differentiate both sides of the Charles' Law equation with respect to time, \(t\). So we will find \(\frac{d}{dt}(V) = \frac{d}{dt}(kT)\).

Apply the derivative rules

Using the derivative rules, we find: - For the left side, the derivative of volume is \(\frac{dV}{dt}\) - For the right side, since \(k\) is a constant, it can be treated as a coefficient. We have \(\frac{d}{dt}(kT) = k\frac{dT}{dt}\)

Write the final equation

Equating the derivatives of both sides, we get: \(\frac{dV}{dt} = k\frac{dT}{dt}\) This equation relates the rate of change of volume, \(\frac{dV}{dt}\), to the rate of change of temperature, \(\frac{dT}{dt}\).

Key concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with instantaneous rates of change and slopes of curves. It allows us to determine how one quantity changes in relation to another. This type of calculus is essential in analyzing motion, optimizing functions, and understanding changes within various physical systems. The exercise in question employs differential calculus to explore how the volume of gas changes as its temperature changes.

In our Charles' Law example, we have the volume, represented by the variable 'V,' and the temperature, designated as 'T.' The law itself establishes a direct proportionality between these two when a gas is at constant pressure. To explore the relationship between the changes in volume and temperature over time requires differentiation, which in this case means applying derivative rules to the equation representing Charles' Law. A derivative represents the rate at which a function is changing at any given point, and for this gas law, it translates to understanding how small increments in temperature impact the volume of the gas.

Gas Laws

Gas laws are fundamental principles in chemistry and physics that describe the behavior of gases under various conditions. Charles' Law, in particular, is pivotal in understanding how a gas's volume and temperature are related when the pressure is held constant. In essence, Charles' Law states that if pressure remains unchanged, the volume of a gas is directly proportional to its absolute temperature (measured in Kelvins).

This relationship is represented mathematically by the equation V = kT, where V is the volume of the gas, T is the absolute temperature, and k is a proportionality constant unique to each sample of gas. When studying how these variables change with respect to time, the relevance of differential calculus emerges. In the context of the explained exercise, understanding this gas law and its implications on the behavior of gases under dynamic conditions is crucial.

Rate of Change

The rate of change in a particular context is a measure of how a quantity changes over time. It's a concept often described as a speed or velocity at which a variable quantity is changing. In the case of Charles' Law, the rate of change helps us quantify how quickly the volume of a gas responds to shifts in temperature over time.

In the solution to our exercise, the rate of change of the volume with respect to time is given by the derivative dV/dt, and similarly, the rate of change of temperature with respect to time is given by dT/dt. By differentiating both sides of the equation with respect to time, we can directly relate these two rates of change, which then describes how the volume of a gas expands or contracts in response to temperature changes when pressure is constant. Understanding this relationship is fundamental for applications in various scientific fields, from meteorology to engineering.