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 "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 0.448 J/g°C.

Step-by-step solution

Understand The Problem

We need to determine the specific heat capacity (c) of a chromium metal piece using the given mass, initial and final temperatures, as well as the relationship with the surrounding water. We know the mass and temperature changes for both chromium and water, and we'll set up an energy balance since heat lost by chromium is gained by water.

Write The Heat Transfer Equation

The heat absorbed by water is equal to the heat released by the chromium metal, which can be expressed by the equation: \[ m_{ ext{water}}c_{ ext{water}}(T_{f, ext{water}} - T_{i, ext{water}}) = m_{ ext{Cr}}c_{ ext{Cr}}(T_{i, ext{Cr}} - T_{f, ext{Cr}}) \]where:- \( m \) is mass,- \( c \) is specific heat capacity,- \( T_i \) and \( T_f \) are initial and final temperatures.

Substitute Known Values

Plug in the known values into the equation:- Mass of water \( m_{ ext{water}} = 82.3 \, ext{g} \)- Initial temperature of water \( T_{i, ext{water}} = 23.3^{ ext{o}}C \)- Final temperature \( T_{f} = 25.6^{ ext{o}}C \)- Specific heat capacity of water \( c_{ ext{water}} = 4.18 \, ext{J/g}^{ ext{o}}C \)- Mass of chromium \( m_{ ext{Cr}} = 24.26 \, ext{g} \)- Initial temperature of chromium \( T_{i, ext{Cr}} = 98.3^{ ext{o}}C \)Fill into the equation:\[ 82.3 imes 4.18 imes (25.6 - 23.3) = 24.26 imes c_{ ext{Cr}} imes (98.3 - 25.6) \]

Simplify and Solve The Equation

Calculate the left-hand side:\[ 82.3 imes 4.18 imes 2.3 = 790.7746 \, ext{J} \]Substitute this into the equation:\[ 790.7746 = 24.26 imes c_{ ext{Cr}} imes 72.7 \]Solve for \( c_{ ext{Cr}} \):\[ c_{ ext{Cr}} = \frac{790.7746}{24.26 imes 72.7} \]

Calculate the Result

Perform the division to find the specific heat capacity:\[ c_{ ext{Cr}} = \frac{790.7746}{1763.902} \approx 0.448 \, ext{J/g}^{ ext{o}}C \]

Write the Conclusion

The specific heat capacity of the chromium metal is approximately \( 0.448 \, ext{J/g}^{ ext{o}}C \).

Key concepts

Calorimetry Basics

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It's all about the transfer of thermal energy. Here, we use a calorimeter, a device designed to measure this heat transfer by observing the change in temperature of a known substance. In this exercise, the object of focus is a piece of chromium metal, and the environment it's interacting with is water within a coffee-cup calorimeter. This system helps us quantify how chromium and water exchange energy.

• Calorimeters measure the heat exchanged between materials when they reach thermal equilibrium.
• They are crucial in determining specific heat capacities, energy transfer in reactions, and more.
• This exercise utilizes a simple coffee-cup calorimeter for practicality and ease of measurement.

By carefully assessing the temperature changes in both the chromium and water, we can determine the amount of heat transferred. This concept is essential in understanding how energy moves between objects in contact with different temperatures.

Understanding Heat Transfer

Heat transfer is the movement of thermal energy from a substance of higher temperature to one of lower temperature. This fundamental principle ensures that all bodies eventually reach thermal equilibrium, where no net heat flows between them. In our exercise, the initially hot chromium transfers heat to the cooler water until both reach the same temperature.

• Heat transfer occurs until thermal equilibrium is achieved—meaning both the chromium and water will settle at the same final temperature.
• Knowing the mass and specific heat capacity of the water helps us calculate the amount of heat it absorbs from the chromium.
• Using the heat transfer equation, we see this energy movement quantified, providing a method for calculating the chromium's specific heat capacity.

This process is highlighted by changes in temperature, with heat leaving the chromium and entering the water as they approach equilibrium together.

Energy Balance Equation

The concept of energy balance is essential to understand how energy is conserved when two bodies interact thermally. In calorimetry, this principle is reflected in the heat transfer equation, where the heat lost by one substance is equal to the heat gained by another.

• In our example, the heat lost by the chromium is exactly equal to the heat gained by the water, assuming no heat loss to the surroundings.
• The energy balance equation reflects this: \[ m_{ \text{water}}c_{ \text{water}}(T_{f, \text{water}} - T_{i, \text{water}}) = m_{ \text{Cr}}c_{ \text{Cr}}(T_{i, \text{Cr}} - T_{f, \text{Cr}}) \]
• This equation helps us isolate the unknown specific heat capacity of the chromium, allowing us to solve it mathematically.

Consistent application of the energy balance principle ensures accurate interpretations of thermal interactions, like the one between our chromium sample and the water. By understanding this balance, we can explore how the specific heat capacities of materials determine their thermal behavior.