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 0.137 J/(g°C), and the molar heat capacity is 27.48 J/(mol°C).

Step-by-step solution

Identify the given information

We are given the following information: Q = 585 J m = 125.6 g T1 = 20.0°C T2 = 53.5°C M (molar mass of mercury) = 200.59 g/mol

Calculate the temperature change (ΔT)

To find ΔT, subtract the initial temperature (T1) from the final temperature (T2): ΔT = T2 - T1 ΔT = 53.5°C - 20.0°C ΔT = 33.5°C

Calculate the specific heat capacity (c)

Use the formula Q = mcΔT to find the specific heat capacity (c): 585 J = (125.6 g)(c)(33.5°C) c = 585 J / (125.6 g × 33.5°C) c = 0.137 J/(g°C)

Calculate the moles of mercury

To find the moles (n) of mercury, divide the mass (m) by its molar mass (M): n = m / M n = 125.6 g / 200.59 g/mol n = 0.626 mol

Calculate the molar heat capacity (C)

Now that we have the specific heat capacity (c) and the moles of mercury (n), we can calculate the molar heat capacity (C) using the formula: C = c × M C = 0.137 J/(g°C) × 200.59 g/mol C = 27.48 J/(mol°C) So, the specific heat capacity of mercury is 0.137 J/(g°C), and the molar heat capacity is 27.48 J/(mol°C).

Key concepts

Molar Heat Capacity

The molar heat capacity of a substance is an important concept in thermodynamics. It is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. Molar heat capacity provides insight into how different materials absorb and store heat energy. In the context of mercury, a metal with a molar mass of 200.59 g/mol, the calculation gives us a specific value that is characteristic of the material's thermal properties.

Molar heat capacity is depicted by the symbol \(C\) and its unit is generally in \(\text{J/(mol°C)}\). By understanding molar heat capacity, we can predict and control thermal processes in practical applications such as thermometers and barometers, where mercury is frequently used due to its predictable thermal behavior.

Energy Calculation

Calculating the energy required for a temperature change involves understanding the relationship between energy, mass, specific heat capacity, and temperature change. The formula used is:

\[ Q = mc\Delta T \]

Here, \(Q\) represents the heat energy in joules, \(m\) is the mass in grams, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change in Celsius.

In our exercise, we know the energy (\(Q\)) is 585 J, which helps us determine the specific heat capacity once the change in temperature and mass are given. This equation shows the proportional relationship between energy and temperature change, reinforcing the idea that more energy results in greater temperature shifts, provided the mass and specific heat capacity remain constant.

Temperature Change

Temperature change, denoted as \(\Delta T\), is a simple yet essential aspect of thermal calculations. It is determined by subtracting the initial temperature from the final temperature.

In the mercury example, the initial temperature \((T_1)\) is 20.0°C and the final temperature \((T_2)\) is 53.5°C. Therefore, the temperature change is calculated as:

\[ \Delta T = T_2 - T_1 = 53.5°C - 20.0°C = 33.5°C \]

This 33.5°C change provides crucial information for calculating other thermal properties, such as specific heat capacity, and allows us to precisely determine how much energy was involved in the temperature shift.

Mercury Properties

Mercury is a unique metal with distinctive thermal and physical properties. Known for its use in thermometers, mercury remains liquid at room temperature and has a high density. These characteristics make it ideal for applications that require precise temperature measurements.

The specific heat capacity of mercury, found to be 0.137 \(\text{J/(g°C)}\), indicates how much energy mercury requires to change its temperature. Meanwhile, its molar heat capacity at 27.48 \(\text{J/(mol°C)}\) further exemplifies its efficiency in energy absorption relative to its molar mass.

Understanding mercury's properties and how they influence its behavior under thermal stress is crucial in both scientific research and practical applications, allowing safe and effective usage of this fascinating element.