Problem 100 - University Physics with Modern Physics

Question

The molar heat capacity of a certain substance varies with temperature according to the empirical equation $$C=29.5 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}+\left(8.20 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2}\right) \mathrm{T}$$ How much heat is necessary to change the temperature of 3.00 mol of this substance from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\) ? (Hint: Use Eq. (17.18) in the form \(d Q=n C d T\) and integrate. \()\)

Step-by-Step Solution

Answer

Integrate the heat equation from \(T_1\) to \(T_2\).

Step 1: Understand the problem

We need to find the amount of heat required to change the temperature of 3.00 mol of a substance from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\). The molar heat capacity \(C(T)\) is given as a function of temperature: \(C(T) = 29.5 + (8.20 \times 10^{-3})T\). To find \(Q\), we integrate \(dQ = nC(T) dT\) over the temperature range.

Step 2: Set up the integration

The formula to use is \(dQ = nC(T) dT\), where \(n\) is the number of moles, \(C(T) = 29.5 + (8.20 \times 10^{-3})T\), and \(dT\) is the differential temperature change. For 3.00 mol, the expression becomes \(dQ = 3.00 (29.5 + (8.20 \times 10^{-3})T) dT\). We need to integrate this from \(T_1 = 27^{\circ}C\) to \(T_2 = 227^{\circ}C\).

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. It looks at how energy is transferred and transformed. In the context of this problem, we are interested in energy transfer in the form of heat, which affects the temperature of a substance. Thermodynamics helps us understand how heated systems reach equilibrium, how energy is conserved, and how it flows from one part of a system to another. Some key concepts in thermodynamics include:

• System and Surroundings: A system is the part of the universe we are focusing on. Everything else is considered the surroundings.
• State Variables: Properties like temperature, pressure, and volume that describe the state of a system.
• Processes: Paths taking a system from one state to another, affecting heat, work, and internal energy.

Understanding these concepts helps us apply them to solve problems involving heat changes.

Temperature Change

Temperature change refers to the amount by which the temperature of a substance increases or decreases. This exercise asks us to calculate the heat required for a certain temperature change using the molar heat capacity of the substance involved. When we talk about temperature changes, we often refer to:

• Initial and Final Temperatures: These define the starting and ending points of our calculations. Here, they are 27°C and 227°C.
• Temperature Difference (ΔT): Calculated as the final temperature minus the initial temperature. In this exercise, ΔT is 200°C.

Recognizing these changes helps us understand how much heat transfer occurs, as the amount of heat required depends on the temperature change, the amount of substance, and its heat capacity.

Heat Transfer

Heat transfer involves the movement of thermal energy from one object or substance to another due to temperature difference. This can occur in various modes: conduction, convection, and radiation.In our exercise, we're concerned with heat transfer needed to change the temperature of a substance. This involves calculating the total heat transferred using the equation:\[ dQ = nC(T) dT \]where:

• dQ: The infinitesimal amount of heat transferred.
• n: Number of moles of the substance, here 3.00 moles.
• C(T): Molar heat capacity, given as a function of temperature.
• dT: Infinitesimal temperature change.

To find the total heat (Q), we integrate this expression over the temperature range of interest. This tells us how much energy is transferred to the substance as heat, changing its temperature from 27°C to 227°C.

Empirical Equation

An empirical equation is a formula derived from experimental data. It is used to describe a relationship observed in nature without necessarily explaining the underlying reasons. In this problem, the molar heat capacity is given as:\[ C(T) = 29.5 + (8.20 \times 10^{-3})T \]This expression shows that the heat capacity increases linearly with temperature. Here, "29.5" is the base capacity, and "8.20 \times 10^{-3}" is the rate at which it changes with temperature.Why use empirical equations?

• Predictions: They help predict how systems behave when we change variables like temperature.
• Simplification: Allow us to model complex behaviors without knowing every detail.
• Practicality: Derived from data, they are often more accessible than theoretical equations.

For this exercise, the empirical equation is essential in integrating the heat capacity to find the total heat required to raise the temperature of the substance.