Question

Mercury was once used in thermometers and barometers. When \(46.9 \mathrm{~J}\) of heat are absorbed by \(100.0 \mathrm{~g}\) of mercury at \(25.00^{\circ} \mathrm{C},\) the temperature increases to \(28.35^{\circ} \mathrm{C}\). What is the specific heat of mercury?

Short answer

Answer: The specific heat of mercury is approximately \(0.140\, \frac{\mathrm{J}}{\text{g}\cdot ^\circ\mathrm{C}}\).

Step-by-step solution

List the given information

We are given the following information: - Mass of mercury: \(m = 100.0\,\text{g}\) - Initial temperature: \(T_i = 25.00^\circ\mathrm{C}\) - Final temperature: \(T_f = 28.35^\circ\mathrm{C}\) - Heat absorbed: \(q = 46.9\,\mathrm{J}\)

Determine the change in temperature

We will need to determine the change in temperature, which is the difference between the final temperature and the initial temperature. We can find this by using the formula: \(\Delta T = T_f - T_i\) \(\Delta T = 28.35^\circ\mathrm{C} - 25.00^\circ\mathrm{C} = 3.35^\circ\mathrm{C}\)

Use the heat equation to solve for the specific heat of mercury

We will now use the heat equation, \(q = mc\Delta T\), and plug in the the values we know to solve for the specific heat of mercury, \(c\). The equation will be rearranged as follows: \(c = \frac{q}{m\Delta T}\). Plug in the values into the equation: \(c = \frac{46.9\,\mathrm{J}}{100.0\,\text{g} \times 3.35^\circ\mathrm{C}}\)

Calculate the specific heat of mercury

Calculate the specific heat of mercury using the values we have plugged into the equation: \(c = \frac{46.9\,\mathrm{J}}{100.0\,\text{g} \times 3.35^\circ\mathrm{C}} \approx 0.140\, \frac{\mathrm{J}}{\text{g}\cdot ^\circ\mathrm{C}}\)

Write the final answer

The specific heat of mercury is approximately \(0.140\, \frac{\mathrm{J}}{\text{g}\cdot ^\circ\mathrm{C}}\).

Key concepts

Heat Transfer

Heat transfer is a key concept in thermodynamics and understanding how substances change temperature. It involves the movement of thermal energy from one object or substance to another when they are at different temperatures. This process continues until thermal equilibrium is reached, meaning the objects involved achieve the same temperature.

In the context of chemistry, different substances have varying abilities to absorb or release heat, which is quantitatively expressed as their specific heat capacity. This is the amount of heat energy required to raise the temperature of a unit mass of the substance by one degree Celsius (or Kelvin). Higher specific heat capacities mean that a substance will absorb more heat to increase its temperature, making it an important factor in understanding temperature changes during chemical reactions or physical processes.

When learning about heat transfer, we consider three primary methods: conduction, which is the direct transfer of heat through a material; convection, the movement of heat through fluids (liquids or gases); and radiation, which is the transfer of heat through electromagnetic waves. These mechanisms underpin many processes in both scientific studies and everyday life.

Temperature Change in Chemistry

Temperature change in chemistry is closely related to the concept of heat transfer. The temperature of a substance changes as it absorbs or releases heat, a process that is central to chemical reactions and phases changes. For example, heating a liquid can cause it to evaporate, while cooling a gas may lead to its condensation.

The specific heat capacity contributes significantly to how a substance responds to heat. Substances with a high specific heat capacity, like water, require more energy to increase their temperature, which is why water heats up and cools down slowly. Conversely, substances such as metals generally have lower specific heat capacities, allowing them to heat up and cool down more quickly.

The relationship between heat added or removed and the resulting temperature change is given by the equation: \(q = mc\Delta T\), where:\

• \(q\) is the quantity of heat in joules (J)
• \(m\) is the mass of the substance in grams (g)
• \(c\) is the specific heat capacity (\(\frac{J}{g\cdot ^\circ C}\))
• \(\Delta T\) is the change in temperature in degrees Celsius (\(^\circ C\))

Using this equation, we can calculate how much energy is needed to raise the temperature of a given mass of a substance to a desired level, or vice versa.

Heat Equation Thermodynamics

The heat equation in thermodynamics, which is central to solving problems involving temperature changes and heat transfer, is \(q = mc\Delta T\). It mathematically describes the relationship between the heat added or removed from a system and the subsequent temperature change, factoring in the specific heat capacity.

The specific heat capacity \(c\) is a crucial part of this equation, as it quantifies the thermal inertia of a substance—the resistance of the substance to temperature change. Higher \(c\) values indicate that more heat is needed to raise the temperature of the substance by one degree. The value of \(c\) can be experimentally determined by measuring the amount of heat energy \(q\) absorbed or released by a substance, along with the corresponding temperature change \(\Delta T\) and the mass \(m\) of the substance.

Revisiting the mercury problem in our exercise, by using the heat equation, we can determine mercury's specific heat capacity based on the amount of heat energy added and the observed temperature change. This piece of information further helps us understand how mercury, as a material, reacts to heating, which is important in applications such as thermometers or barometers as well as in industrial processes. The precise calculation using the equation ensures accuracy in assessing how much a temperature will change for a known quantity of heat energy, a vital aspect in chemistry and thermodynamics.