Question

For the ideal gaseous reaction, the rate is generally expressed in terms of \(\frac{d P}{d t}\) instead of \(\frac{d C}{d t}\) or \(\frac{d n}{d t}\) (where \(C=\frac{n}{V}\) is concentration and \(n\) the no. of moles). What is the reaction among these three expressions if \(T\) and \(V\) are constant? (a) \(\frac{d C}{d t}=\frac{d n}{d t}=\frac{d P}{d t}\) (b) \(\frac{d C}{d t}=\frac{1}{V} \frac{d n}{d t}=\frac{1}{R T}\left(\frac{d P}{d t}\right)\) (c) \(R T \frac{d C}{d t}=\frac{d n}{d t}=\frac{d P}{d t}\) (d) None of these

Short answer

The relation is given by (b) \(\frac{dC}{dt} = \frac{1}{V} \frac{dn}{dt} = \frac{1}{RT} \left(\frac{dP}{dt}\right)\).

Step-by-step solution

Relate the Change in Concentration to the Change in Moles

Using the definition of concentration, where \(C = \frac{n}{V}\), differentiate both sides with respect to time while keeping the volume constant to find \(\frac{dC}{dt}\). Since \(V\) is constant, applying the derivative yields \(\frac{dC}{dt} = \frac{1}{V} \frac{dn}{dt}\).

Relate the Change in Pressure to the Change in Moles Using Ideal Gas Law

The Ideal Gas Law is \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(T\) is temperature, \(n\) is moles, and \(R\) is the universal gas constant. Differentiate both sides of this equation with respect to time under constant temperature and volume to obtain the rate of change of pressure in terms of the rate of change of moles: \(V\frac{dP}{dt} = R\frac{dT}{dt} + RT\frac{dn}{dt}\). Since \(T\) and \(V\) are constant, their derivatives equal zero, simplifying the equation to \(V\frac{dP}{dt} = RT\frac{dn}{dt}\), or \(\frac{dP}{dt} = \frac{RT}{V}\frac{dn}{dt}\).

Combine the Equations to Find the Relationship

From steps 1 and 2, we can combine the relationships \(\frac{dC}{dt} = \frac{1}{V}\frac{dn}{dt}\) and \(\frac{dP}{dt} = \frac{RT}{V}\frac{dn}{dt}\). Solving for \(\frac{dn}{dt}\) from the first relation and substituting into the second yields \(\frac{dP}{dt} = \frac{RT}{V}(V\frac{dC}{dt}) = RT\frac{dC}{dt}\). Dividing both sides by \(RT\) to isolate \(\frac{dC}{dt}\) we get \(\frac{dC}{dt} = \frac{1}{V}\frac{dn}{dt} = \frac{1}{RT} \frac{dP}{dt}\), which matches option (b).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a cornerstone of physical chemistry, providing a relationship between the pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. The law is often expressed as the equation \( PV = nRT \), where R represents the universal gas constant. This equation assumes that the gas particles are in random motion and that they do not interact with each other.

Understanding the Ideal Gas Law is crucial when dealing with gases under various conditions of temperature and pressure. It allows chemists to predict how a gas will behave when it is compressed, heated, or mixed with other gases. For example, if you have a sealed container of gas and you know three of the four variables (P, V, T, or n), you can calculate the fourth. This is particularly relevant in our exercise, as the relationship between the change in pressure and the number of moles is derived from differentiating the Ideal Gas Law with respect to time under constant volume and temperature.

Rate of Change in Chemistry

In chemical kinetics, the rate of change is a measure of how the concentration of a substance changes over time. It's often denoted as \( \frac{dC}{dt} \) for concentration or \( \frac{dP}{dt} \) for pressure in the case of gases. The rate can indicate how quickly a reactant is consumed or a product is formed in a chemical reaction. It's fundamental for understanding reaction mechanisms, predicting reaction outcomes, and designing industrial processes.

To calculate the rate of change in a gaseous system where the volume is constant, one can utilize the relationships provided by the Ideal Gas Law. In chemistry, these computations are vital for predicting how conditions such as pressure and volume affect reaction rates. As applied to our exercise, by keeping T and V constant and knowing the rate of change of either moles \( \frac{dn}{dt} \) or pressure \( \frac{dP}{dt} \), we can deduce the rate of change in concentration.

Differentiation in Physical Chemistry

Differentiation is a mathematical operation used to calculate the rate at which a quantity changes and it is widely applied in physical chemistry to determine reaction rates and changes in system properties over time. By differentiating an equation, chemists can find expressions that explain how variables such as pressure and concentration evolve.

In the context of our exercise, differentiation helps in understanding how the rate of change of concentration (\( \frac{dC}{dt} \)) relates to the rate of change of moles (\( \frac{dn}{dt} \)) and pressure (\( \frac{dP}{dt} \)). The step-by-step solution demonstrates the application of differentiation to the Ideal Gas Law under constant temperature and volume conditions, leading to an expression that relates these rates. This method is essential for exploring the dynamic aspects of chemical reactions and the conditions that influence them.