Question

A sample of nickel is heated to \(99.8^{\circ} \mathrm{C}\) and placed in a coffeecup calorimeter containing \(150.0 \mathrm{~g}\) water at \(23.5^{\circ} \mathrm{C}\). After the metal cools, the final temperature of metal and water mixture is \(25.0^{\circ} \mathrm{C}\). If the specific heat capacity of nickel is \(0.444 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\), what mass of nickel was originally heated? Assume no heat loss to the surroundings.

Short answer

The mass of nickel that was originally heated is approximately \(33.5 \text{~g}\).

Step-by-step solution

Write down the equations for heat gained and heat lost

Since the heat gained by water is equal to the heat lost by the nickel, we can write the following equation: \(q_\text{gained, water} = q_\text{lost, nickel}\) \(mc_\text{water}\Delta T_\text{water} = mc_\text{nickel}\Delta T_\text{nickel}\)

Substitute the given values for water and nickel

We have the following information given: \(m_\text{water} = 150.0 \mathrm{~g}\) \(T_{\text{initial, water}} = 23.5^{\circ}\mathrm{C}\) \(T_{\text{final}} = 25.0^{\circ}\mathrm{C}\) \(c_\text{water} = 4.18 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g}\) (specific heat capacity of water) \(T_{\text{initial, nickel}} = 99.8^{\circ}\mathrm{C}\) \(c_\text{nickel} = 0.444 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g}\) From the information above, we can find the ΔT's for both water and nickel: \(\Delta T_\text{water} = T_{\text{final}} - T_{\text{initial, water}}\) \(\Delta T_\text{nickel} = T_{\text{final}} - T_{\text{initial, nickel}}\)

Calculate the final ΔT's for water and nickel

Now substitute the values for the initial and final temperatures to find ΔT's for water and nickel: \(\Delta T_\text{water} = 25.0^{\circ}\mathrm{C} - 23.5^{\circ}\mathrm{C} = 1.5^{\circ}\mathrm{C}\) \(\Delta T_\text{nickel} = 25.0^{\circ}\mathrm{C} - 99.8^{\circ}\mathrm{C} = -74.8^{\circ}\mathrm{C}\)

Solve for the mass of nickel in the equation

The complete equation is: \(m_\text{water}c_\text{water}\Delta T_\text{water} = m_\text{nickel}c_\text{nickel}\Delta T_\text{nickel}\) Now substitute the known values: \(150.0 \mathrm{~g} \times 4.18 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g} \times 1.5^{\circ}\mathrm{C}= m_\text{nickel}\times 0.444 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g}\times-74.8^{\circ}\mathrm{C}\) And now we can solve for the mass of nickel: \(m_\text{nickel} = \frac{150.0 \mathrm{~g} \times 4.18 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g} \times 1.5^{\circ}\mathrm{C}}{0.444 \mathrm{~J} / {~}^{\circ}\mathrm{C} \cdot \mathrm{g} \times -74.8^{\circ}\mathrm{C}}\)

Calculate the mass of nickel

Now do the calculation using the values above: \(m_\text{nickel} = \frac{939.3\text{~J}}{(-74.8^{\circ}\mathrm{C}\times0.444\,\text{J/}^{\circ}\mathrm{C}\,\text{g})}\) \(m_\text{nickel} = 33.5\,\text{g}\) So, the mass of nickel that was originally heated is approximately 33.5 grams.

Key concepts

Heat Transfer Calculations

Heat transfer calculations are fundamental to understanding how energy in the form of heat moves from one body to another. These calculations are essential in calorimetry, a process used to measure the amount of heat involved in chemical reactions or physical changes. To perform such calculations, we use the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred.

For instance, when a hot object is placed in cooler water, heat will transfer from the object to the water until thermal equilibrium is reached – this is the moment when both the object and the water have the same temperature. By setting the heat lost by the hot object equal to the heat gained by the cooler water, we can form an equation that allows us to solve for unknown values, such as the mass of the object, given we have the specific heat capacities and the temperature changes for both substances.

Specific Heat Capacity

The specific heat capacity (often simply called specific heat) is the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius. Specific heat is a property intrinsic to the material and varies from one material to another. The concept of specific heat is crucial in calorimetry problems because it allows us to quantify the heat transfer that occurs during temperature changes.

In our exercise, nickel and water have different specific heat capacities, which means they require different amounts of heat energy to experience the same change in temperature. By knowing these capacities, we can establish a relation between the heat absorbed or released and the corresponding temperature change, allowing us to determine unknown values such as the mass of the nickel or the final temperature of the system.

Thermal Equilibrium

Thermal equilibrium refers to the state in which two or more objects in contact with each other no longer transfer heat, meaning they have reached the same temperature. In the context of calorimetry problems, reaching thermal equilibrium is the endpoint we consider for our calculations. Once the system has achieved thermal equilibrium, we apply the formula that equates the heat lost by one substance to the heat gained by the other. It is important to note that in a perfectly insulated system, such as our assumed coffee cup calorimeter, the heat exchange occurs exclusively between the substances in contact, without any loss to the surroundings.

Understanding thermal equilibrium is vital for solving calorimetry problems since the final temperature of the system provides a benchmark for determining the direction and magnitude of heat transfer. This is essential for calculating parameters like the mass or specific heat capacity when they are unknown.