Question

Assume that it takes \(0.0700 \mathrm{~J}\) of energy to heat a \(1.00-\mathrm{g}\) sample of mercury from \(10.000^{\circ} \mathrm{C}\) to \(10.500{ }^{\circ} \mathrm{C}\) and that the heat capacity of mercury is constant, with a negligible change in volume as a function of temperature. Find the change in entropy if this sample is heated from \(10 .{ }^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\).

Short answer

Question: Calculate the change in entropy of a 1.00-g sample of mercury when it is heated from 10.0°C to 100.0°C, given that it takes 0.0700 J of energy to heat the sample from 10.0°C to 10.5°C and assuming a constant heat capacity. Answer: The change in entropy of the mercury sample when heated from 10.0°C to 100.0°C is approximately 0.175 J K⁻¹.

Step-by-step solution

Determine the heat capacity of mercury

Given that it takes 0.0700 J of energy to heat a 1.00-g sample of mercury from 10.0°C to 10.5°C, we can determine the specific heat capacity (c) of mercury using the formula: q = mcΔT Where: q = heat energy (0.0700 J), m = mass of the sample (1.00 g), c = specific heat capacity, and ΔT = change in temperature (0.5 K). Rearranging the formula for c, we get: c = q/(mΔT) Substituting the given values, we find the specific heat capacity of mercury: c = 0.0700 J / (1.00 g * 0.5 K) = 0.140 J g⁻¹ K⁻¹

Calculate the change in entropy

To find the change in entropy (ΔS) for heating the mercury sample from 10.0°C to 100.0°C, we use the following formula: ΔS = mc\int_{T_1}^{T_2} \frac{dT}{T} Where: ΔS = change in entropy, m = mass of the sample (1.00 g), c = specific heat capacity (0.140 J g⁻¹ K⁻¹), T_1 = initial temperature (10.0°C = 283.15 K), and T_2 = final temperature (100.0°C = 373.15 K), In this case, c is constant, so we can take it out of the integral: ΔS = mc (ln(T_2) - ln(T_1)) Substituting the values we found, we can now calculate the change in entropy: ΔS = (1.00 g)(0.140 J g⁻¹ K⁻¹) (ln(373.15 K) - ln(283.15 K)) ΔS ≈ 0.140 J K⁻¹ (1.2507) = 0.175 J K⁻¹ The change in entropy of the mercury sample when heated from 10.0°C to 100.0°C is approximately 0.175 J K⁻¹.

Key concepts

Specific Heat Capacity

Understanding the concept of specific heat capacity is crucial for solving problems involving temperature changes and energy transfer. Specific heat capacity, often just called specific heat, is defined as the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius (or one kelvin).

It is an intrinsic property of a substance, meaning it doesn't change with the amount of substance you have; whether you have 1 gram or 1000 grams, the specific heat capacity will be the same for a given substance under consistent conditions. To grasp this idea, imagine heating up different materials, like water and metal. You might have noticed that metal heats up faster compared to water. This is because metals typically have a lower specific heat capacity than water, requiring less energy to raise their temperature.

In thermodynamics, the specific heat capacity is a gateway to understanding energy transfer. When applied to the dimension of entropy change calculation, specific heat tells us how much energy is dispersed or drawn in during a temperature change, which we interpret as a measure of disorder or randomness in the system.

Thermodynamics

Thermodynamics is a branch of physics that studies the movement of heat and its conversion to different forms of energy. At its heart are the laws of thermodynamics which govern these energy transformations. One fundamental concept here is entropy, often described as the measure of the disorder or randomness in a system. The second law of thermodynamics states that for any isolated system, entropy can only increase, explaining why processes are irreversible and why we can't completely convert all heat energy to useful work.

In the context of our exercise, thermodynamics helps us understand how heat energy relates to changes in entropy. When heat is added to a system, like our mercury sample, it tends to disperse the energy throughout the system, increasing its overall entropy. Calculating this change provides insight into the 'spread' of energy and helps decode the complexity of thermal processes.

Integral Calculus in Physics

In the world of physics, integral calculus is a powerful tool used to calculate quantities that involve summation over continuous variables. Integrals, for instance, can tell us the total energy work done along a path or the change in a physical quantity over time or space. When it comes to thermodynamics, integrals become crucial in calculating the total entropy change over a range of temperatures as we've seen in the textbook exercise.

To find the change in entropy, we performed an integral over the temperature range with the formula involving the natural logarithm function. This approach is made possible by integral calculus and allows us to summarize the continuous increase in entropy at each infinitesimal step as the temperature rises from the initial to the final state. It's the integral's calculation that yields the total change in entropy in a neat, quantifiable number, reflecting the cumulative effect of the heat transfer.