Question

Assume that it takes 0.0700 J of energy to heat a \(1.00-g\) 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 \(10.0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\).

Short answer

Answer: The change in entropy when heating the 1.00-g sample of mercury from 10.0°C to 100°C is approximately 1.952 J/K.

Step-by-step solution

Calculate the specific heat capacity of mercury

The formula to calculate the specific heat capacity of a substance is: c = (Q / mΔT) where: c = specific heat capacity (J/g°C) Q = energy (J) m = mass of the substance (g) ΔT = change in temperature (°C) In our case, we are given: Q = 0.0700 J m = 1.00 g ΔT = 10.500 - 10.000 = 0.500°C Now, let's calculate c: c = (0.0700 J) / (1.00 g * 0.500°C) = 0.140 J/g°C

Calculate the heat needed to raise the temperature from 10.0°C to 100°C

Now, we can use the specific heat capacity to find the heat needed to raise the temperature of the 1.00-g sample from 10.0°C to 100°C. ΔT_new = 100°C - 10.0°C = 90.0°C Using the formula Q = m * c * ΔT: Q_new = 1.00 g * 0.140 J/g°C * 90.0°C = 12.6 J

Calculate the entropy change

The change in entropy (ΔS) is given by the formula: ΔS = Q / T However, since the temperature is changing, we need to integrate over the temperature range. The formula becomes: ΔS = ∫ (Q / T) dT Since the heat capacity is constant, Q_new can be written in terms of c: Q_new = c * mΔT_new Thus, ΔS can be written as: ΔS = ∫ (c * m / T) dT Now, we can integrate this equation over the temperature range (10°C to 100°C) ΔS = m * c * ln(T_final / T_initial) Given that this is an isochoric process (constant volume), the entropy change can be calculated using: ΔS = 1.00 g * 0.140 J/g°C * ln(373K / 283K) = 1.952 J/K The change in entropy when heating the 1.00-g sample of mercury from 10.0°C to 100°C is approximately 1.952 J/K.

Key concepts

Specific Heat Capacity

Specific heat capacity is a physical property of a substance that tells us how much heat energy is required to raise the temperature of one gram of the substance by one degree Celsius. It is represented by the symbol 'c' and is usually expressed in units of joules per gram per degree Celsius (J/g°C). Understanding specific heat capacity is crucial for solving problems involving heat transfer within thermodynamics.

The formula to find the specific heat capacity is: c = \(\frac{Q}{m\Delta T}\)where 'c' is specific heat capacity, 'Q' stands for the amount of heat added or removed, 'm' is the mass of the object, and \(\Delta T\) represents the change in temperature. Applied to the given problem, we use this formula to calculate the energy required to change the mercury's temperature by 0.5°C, allowing us to understand how mercury will behave as it is heated from 10°C to 100°C.

Specific heat capacity is an essential factor for many applications, including designing heating and cooling systems or understanding the climate's response to solar radiation.

Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. The behavior of these quantities is governed by four laws of thermodynamics, which convey a quantitative description using measurable macroscopic physical quantities. However, classical thermodynamics is mostly concerned with the system in equilibrium states.

In the context of this exercise, we are dealing with the concepts of heat transfer (Q) and entropy change (\(\Delta S\)). Entropy is a measure of the molecular disorder or randomness within a system, and an increase in entropy reflects an increase in this disorder. When we heat the mercury, we are adding energy to the system, which increases the motion of the atoms, thus increasing entropy.

The entropy change for a temperature variation at constant specific heat can be calculated by integrating the amount of heat transferred over the temperature range, assuming a reversible process. This thermodynamic understanding aids in predicting how systems respond to energy changes, important in fields such as meteorology, engineering, and even cosmology.

Integral Calculus in Physics

Integral calculus is a part of mathematics that deals with integrating functions to find quantities like areas, volumes, and, in the case of physics, various physical quantities over a range. It is a fundamental tool in thermodynamics, allowing us to calculate work done, heat transferred, or entropy changes in a system, where the processes are dynamic and often require integration over a variable.

In thermodynamics, when there is a non-constant process, such as a temperature change, the integration becomes necessary. For the mercury sample's entropy change, the formula c = \(\frac{Q_{new}}{T}\) changes to an integral due to the nature of entropy, which integrates the ratio of heat added to temperature over the entire temperature range, with the equation becoming: \(\Delta S = \int \left(\frac{c \cdot m}{T}\right) dT\).This integral is evaluated from the initial to the final temperature, resulting in the natural logarithm of the ratio of these temperatures, multiplied by the specific heat capacity and mass of mercury. This integral part of the calculation shows the power of calculus in physics—transforming the concept of incremental changes into a comprehensive view of the process.