Question

A piece of chromium metal with a mass of \(24.26 \mathrm{g}\) is heated in boiling water to \(98.3^{\circ} \mathrm{C}\) and then dropped into a coffee-cup calorimeter containing \(82.3 \mathrm{g}\) of water at \(23.3^{\circ} \mathrm{C} .\) When thermal equilibrium is reached, the final temperature is \(25.6^{\circ} \mathrm{C} .\) Calculate the specific heat capacity of chromium.

Short answer

The specific heat capacity of chromium is approximately 0.449 J/g°C.

Step-by-step solution

Understanding the Energy Transfer

In this calorimetry problem, energy is transferred from the hot chromium metal to the water until thermal equilibrium is achieved. The heat lost by the metal is equal to the heat gained by the water, which helps us find the specific heat capacity of chromium.

Write the Heat Transfer Equation

The heat gained by the water can be expressed as \( q_{water} = m_{water} \times C_{water} \times \Delta T_{water} \), where \( m_{water} = 82.3 \text{ g} \), \( C_{water} = 4.18 \text{ J/g°C} \), and \( \Delta T_{water} = 25.6°C - 23.3°C = 2.3°C \).

Calculate Heat Gained by Water

Substitute the values into the water's heat equation: \[ q_{water} = 82.3 \times 4.18 \times 2.3 = 793.0426 \text{ J} \]

Write the Heat Transfer Equation for Chromium

The heat lost by chromium can be written as \( q_{chromium} = m_{chromium} \times C_{chromium} \times \Delta T_{chromium} \), where \( m_{chromium} = 24.26 \text{ g} \), and \( \Delta T_{chromium} = 98.3°C - 25.6°C = 72.7°C \).

Set Heat Gained Equal to Heat Lost

Since no heat is lost to the surroundings, \( q_{chromium} = -q_{water} \). Substitute the known values and set them equal:\[ 24.26 \times C_{chromium} \times 72.7 = 793.0426 \]

Solve for Chromium's Specific Heat Capacity

Rearrange the equation to solve for \( C_{chromium} \):\[ C_{chromium} = \frac{793.0426}{24.26 \times 72.7} \approx 0.449 \text{ J/g°C} \]

Key concepts

Specific Heat Capacity

Specific heat capacity is a property of a material that tells us how much heat energy is needed to raise the temperature of one gram of the substance by one degree Celsius. It gives us an idea of how a material absorbs and holds heat.

For instance, water has a high specific heat capacity, which is why it’s so effective at regulating temperatures in nature. In contrast, metals typically have lower specific heat capacities, meaning they heat up and cool down quickly. In the exercise, we're trying to determine the specific heat capacity of chromium.

• This involves determining how much heat energy is transferred when the chromium is placed in water.
• Once the chromium reaches thermal equilibrium, we can calculate the specific heat capacity using the heat transfer equation.

Thermal Equilibrium

Thermal equilibrium is achieved when two objects at different temperatures reach the same temperature after energy is transferred between them. In our problem, the hot chromium metal is placed in cooler water until they both reach the same final temperature of 25.6°C.

This concept is critical in calorimetry experiments, because it allows us to make accurate calculations about heat transfer.

• Knowing the final equilibrium temperature helps us determine how much heat has been transferred.
• It is assumed that no heat was lost to the surroundings in this controlled calorimetry setup.
• This makes it possible to relate the heat lost by the metal directly to the heat gained by the water.

Heat Transfer Equation

The heat transfer equation is central to solving problems in calorimetry and applies the principle of energy conservation. This equation is written as:

• Heat gained by water: \( q_{water} = m_{water} \times C_{water} \times \Delta T_{water} \)
• Heat lost by metal: \( q_{metal} = m_{metal} \times C_{metal} \times \Delta T_{metal} \)

In this exercise, these equations show how energy flows from chromium to the water until equilibrium is reached. We calculate the amount of heat each material gains or loses based on its mass (\( m \)), specific heat capacity (\( C \)), and temperature change (\( \Delta T \)).

The core idea is that the heat lost by the chromium equals the heat gained by the water, which allows us to solve for the unknown specific heat capacity of chromium.

Energy Transfer

Understanding energy transfer is essential for grasping how calorimetry works. Energy transfer occurs as heat flows from a hot object to a cooler one, attempting to reach equilibrium. In calorimetry,

• We measure this energy change to understand the specific heat capacities of materials.
• We assume energy is conserved, meaning it can neither be created nor destroyed, only transferred.

In the exercise, energy from the hot chromium metal is transferred to the cooler water. This transfer continues until both substances reach the same temperature, using up all the energy available for transfer.

This real-world process of energy transfer helps us practice how various substances react to thermal energy and how we can predict their reactions based on scientific principles.