Question

It takes \(585 \mathrm{~J}\) of energy to raise the temperature of \(125.6 \mathrm{~g}\) mercury from \(20.0^{\circ} \mathrm{C}\) to \(53.5^{\circ} \mathrm{C}\). Calculate the specific heat capacity and the molar heat capacity of mercury.

Short answer

The specific heat capacity of mercury is \(0.1384 \frac{J}{g\cdot K}\) and its molar heat capacity is \(28 \frac{J}{mol\cdot K}\).

Step-by-step solution

Recall the formula for heat transfer

The formula for heat transfer (Q) when dealing with temperature changes is given by: \( Q = mc\Delta{T} \) where: - \(Q\) is the heat transferred (in joules), - \(m\) is the mass of the substance (in grams), - \(c\) is the specific heat capacity of the substance (in \( \frac{J}{g\cdot K} \)), - \(\Delta{T} \) is the change in temperature (in Kelvin or Celsius).

Identify the given values

We are given: - The energy needed, \( Q = 585 J \) - The mass of mercury, \( m = 125.6 g \) - The initial temperature, \( T_{1} = 20.0^{\circ}C \) - The final temperature, \( T_{2} = 53.5^{\circ}C \)

Calculate the change in temperature

We need to find the change in temperature, \(\Delta{T}\), which is the difference between the final and initial temperature: \(\Delta{T} = T_{2} - T_{1}\) \(\Delta{T}= 53.5^{\circ}C - 20.0^{\circ}C = 33.5^{\circ}C\)

Calculate the specific heat capacity

Rearrange the heat transfer formula to solve for \(c\): \( c = \frac{Q}{m\Delta{T}} \) Then, plug in the given values and calculate the specific heat capacity: \( c = \frac{585 J}{(125.6 g)(33.5^{\circ}C)}\) \( c = 0.1384 \frac{J}{g\cdot K} \)

Calculate the molar heat capacity

To calculate the molar heat capacity, we need to know the molar mass of mercury. The molar mass of mercury (Hg) is 200.59 g/mol. The molar heat capacity (C) is defined as the product of the specific heat capacity and the molar mass. Use the formula: \( C = c \cdot M \) where: - \(C\) is the molar heat capacity of mercury (in \( \frac{J}{mol\cdot K} \)) - \(c\) is the specific heat capacity of mercury which we calculated in Step 4 (in \( \frac{J}{g\cdot K} \)) - \(M\) is the molar mass of mercury (in g/mol) Plug in the given values and calculate the molar heat capacity: \( C =(0.1384\frac{J}{g\cdot K})(200.59\frac{g}{mol})\) \(C = 27.787 \frac{J}{mol\cdot K}\) Molar heat capacity should have 2 significant figures as there are 2 significant figures in the given values: \(C = 28 \frac{J}{mol\cdot K}\) The specific heat capacity of mercury is \(0.1384 \frac{J}{g\cdot K}\) and its molar heat capacity is \(28 \frac{J}{mol\cdot K}\).

Key concepts

Heat Transfer

Understanding heat transfer is crucial in thermodynamics and various scientific applications. It refers to the movement of thermal energy from one body or substance to another. In our example, heat transfer is measured during the process of increasing the temperature of mercury.

In simple terms, heat transfer occurs when an object or substance absorbs or releases energy, leading to a temperature change. The formula for calculating the amount of heat transferred, known as the heat transfer equation, is: \[\begin{equation} Q = mc\triangle{T} \text{ where: } \begin{itemize} \text{ Q is the heat transferred (in joules), } \text{ m is the mass of the substance (in grams), } \text{ c is the specific heat capacity (in } \frac{J}{g\cdot K}), \text{ }\triangle{T} \text{ is the change in temperature (in Kelvin or Celsius). } \text{ When calculating heat transfer, it's important to keep units consistent and understand that the specific heat capacity, c, is a fundamental property of the substance that indicates how much heat is required to raise the temperature of a unit mass by one degree. } \text{ In educational terms, it's analogous to how much effort a student needs to invest to increase their knowledge by one 'unit' of information—the harder the subject (or the lower the specific heat capacity), the more effort (or heat) is needed for a noticeable improvement. } \text{ } \end{itemize} \text{ } \end{equation}\]

Molar Heat Capacity

Molar heat capacity is a physical property related to the specific heat capacity but with a focus on moles rather than mass. It is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin). This concept is helpful when dealing with substances in chemical reactions where mole-based calculations are more practical.

The molar heat capacity (\(C\)) can be calculated using the specific heat capacity (\(c\)) and the molar mass (\(M\)) of the substance: \[\begin{equation} C = c \cdot M \text{ Here, } \text{ C is the molar heat capacity (in } \frac{J}{mol\cdot K}\text{),} \text{ molar mass M is expressed in grams per mole (g/mol). } \text{ For our mercury example, after determining the specific heat capacity, we use mercury’s molar mass to calculate its molar heat capacity, providing a value that can be a reference for larger-scale or mole-based applications. } \text{ It's essential for students to grasp that while the specific heat capacity relates to the inherent property of the substance's mass, molar heat capacity ties it to the chemical nature of the substance—its molecular structure and mole concept, akin to understanding the nuances of a complex theory, not just in parts but as a whole. } \text{ } \end{equation}\]

Thermochemistry Calculations

Thermochemistry calculations involve quantifying the heat changes that occur during chemical reactions. In the context of our exercise, the calculation focused on the specific heat capacity and molar heat capacity illustrates a simple form of these thermochemical concepts.

To perform these calculations effectively, students must understand the relationship between heat transfer, specific heat capacity, and the molar heat capacity of substances. The process typically involves several steps: identifying the amount of heat transferred (\(Q\)), calculating the temperature change (\(\Delta T\)), and then applying the appropriate formulas to compute the required thermal properties.

When dealing with more complex systems, these calculations can become intricate, involving the assessment of enthalpies of reactions, Hess's law, and the principles of calorimetry. Nevertheless, the foundational calculations as demonstrated in our mercury example set the stage for more advanced thermochemistry applications. Just like solving a complex mathematical equation, thermochemistry builds on basic principles to reach profound conclusions about the energy and heat involved in chemical processes.