Problem 18 - Basic Engineering Mathematics

Question

The molar heat capacity of a solid compound is given by the equation \(c=a+b T\), where \(a\) and \(b\) are constants. When \(c=52, T=100\) and when \(c=172, T=400\). Determine the values of \(a\) and \(b\).

Step-by-Step Solution

Answer

The constants are \(a = 12\) and \(b = 0.4\).

Step 1: Formulate Equations

We are given the formula for molar heat capacity, \(c = a + bT\). Using the provided values, we can establish two equations. The first measurement gives us \(c = 52, T = 100\), forming the equation \(52 = a + 100b\). The second measurement gives us \(c = 172, T = 400\), forming the equation \(172 = a + 400b\).

Step 2: Solve the System of Equations

We have the system of equations: \(52 = a + 100b\) and \(172 = a + 400b\). To eliminate \(a\), subtract the first equation from the second: \(172 - 52 = (a + 400b) - (a + 100b)\). This simplifies to \(120 = 300b\).

Step 3: Solve for Constant \(b\)

From the equation \(120 = 300b\), we isolate \(b\): \(b = \frac{120}{300} = \frac{2}{5}\). So, \(b = 0.4\).

Step 4: Solve for Constant \(a\)

Substitute \(b = 0.4\) back into one of the original equations, say \(52 = a + 100b\): \(52 = a + 100 \times 0.4\), which simplifies to \(52 = a + 40\). Solving for \(a\), we get \(a = 12\).

Key Concepts

System of Equations

Understanding a system of equations is crucial when solving problems involving multiple variables. A system of equations consists of two or more equations with the same set of unknowns. The goal is to find the values of these unknowns that satisfy all equations simultaneously.
In the context of the provided exercise, we generated a system of equations from the molar heat capacity formula. We had two equations:

• \(52 = a + 100b\)
• \(172 = a + 400b\)

These represent the conditions for different temperatures and corresponding heat capacities. To find values for \(a\) and \(b\), we solved these equations together. By aligning and manipulating equations, such as subtracting one from the other, we can often isolate and solve for a variable. This step transforms the system into a simpler format, ultimately allowing for the resolution of each unknown.

Thermodynamics

Thermodynamics is a branch of physical science that studies the movement of heat and how it affects matter. Heat capacity, including molar heat capacity, is a key concept.
Heat capacity refers to the amount of heat required to change a substance's temperature by a certain amount. Molar heat capacity (symbolized often as \(c\)), specifies this heat change per mole of a substance, which brings its role in thermodynamic equations into play. The exercise you've encountered uses a linear equation to express this concept.
The linear nature of the equation \(c = a + bT\) indicates that molar heat capacity changes consistently with temperature. Understanding this linear change is crucial in thermodynamics, as it helps predict how a substance will react in different thermal conditions.

Linear Equation

Linear equations are mathematical expressions that result in a straight line when graphed. They are fundamental in various branches of mathematics and are encountered frequently in real-world problems.
The provided exercise uses the linear equation \(c = a + bT\) to model the relationship between molar heat capacity \(c\) and temperature \(T\). Here, \(a\) and \(b\) are constants representing the y-intercept and slope, respectively. The linearity is evident in the equation because each change in temperature leads to a proportional change in heat capacity.
This understanding is vital because linear equations are straightforward and predictable. Knowing how to manipulate and solve them, as shown in the given problem, allows for the extraction of meaningful scientific insights, such as predicting heat capacity at various temperatures.