Question

Calculate the amount of heat required to heat a \(3.5-\mathrm{kg}\) gold bar from \(21{ }^{\circ} \mathrm{C}\) to \(67{ }^{\circ} \mathrm{C}\).

Short answer

20790 J of heat is needed.

Step-by-step solution

Identify the Specific Heat Value of Gold

The specific heat capacity of gold is required to calculate the heat needed to raise its temperature. This value is typically found in reference materials. For gold, the specific heat is approximately 0.129 J/g°C.

Convert Mass from Kilograms to Grams

Since the specific heat capacity is given in terms of grams, the mass of the gold bar must be converted from kilograms to grams. Multiply the mass by 1000 to convert kilograms to grams: \(3.5 \text{ kg} \times 1000 = 3500 \text{ g}\).

Calculate the Temperature Change

Subtract the initial temperature from the final temperature to find the temperature change: \(67{ }^\circ \mathrm{C} - 21{ }^\circ \mathrm{C} = 46{ }^\circ \mathrm{C}\).

Calculate the Heat Required (Q)

Use the formula \(Q = mc\Delta T\), where \(m\) is the mass in grams, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. Plug in the values calculated: \(Q = 3500 \text{ g} \times 0.129 \text{ J/g}°C \times 46{ }^\circ \mathrm{C}\).

Perform the Calculation

Multiply the mass, specific heat, and temperature change to obtain the heat energy: \(Q = 3500 \times 0.129 \times 46 = 20691 \text{ J}\).

Key concepts

Specific Heat Capacity

Understanding the specific heat capacity is critical when learning about heat calculation in chemistry. It is a property that defines how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius (°C). Different materials have different abilities to store heat, which is why substances like water have a high specific heat capacity while metals like gold have much lower values.

For instance, gold's specific heat capacity is given as 0.129 J/g°C, which is relatively low. This means it takes only 0.129 joules of energy to increase the temperature of one gram of gold by one degree Celsius. In practical terms, the low specific heat capacity indicates that gold will heat up quickly when energy is applied.

Temperature Change

Temperature change, denoted by \(\Delta T\), is an essential part of the heat calculation process. It represents the difference in temperature between the final state and the initial state of a substance. To calculate the temperature change, one must subtract the initial temperature from the final temperature.

In our exercise, the gold bar's temperature change is determined by performing the calculation \(67{ }^\circ \mathrm{C} - 21{ }^\circ \mathrm{C} = 46{ }^\circ \mathrm{C}\). This \(46{ }^\circ \mathrm{C}\) temperature change is what we seek to achieve by adding heat.

Heat Energy Calculation

The process of calculating the amount of heat energy required involves using the concept of specific heat capacity and the intended temperature change. The formula for calculating heat energy \(Q\) is given as \(Q = mc\Delta T\), where \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change.

By utilizing this formula, we can figure out that the heat energy needed to heat our gold bar from \(21{ }^\circ \mathrm{C}\) to \(67{ }^\circ \mathrm{C}\) is \(Q = 3500 \text{ g} \times 0.129 \text{ J/g}\degree C \times 46{ }^\circ \mathrm{C} = 20691 \text{ J}\). This result tells us exactly how much energy we must supply to the gold bar to achieve the desired temperature increase.