Question

How much heat in kilojoules is required to heat \(56 \mathrm{~L}\) of water from an initial temperature of \(85^{\circ} \mathrm{F}\) to a final temperature of \(212{ }^{\circ} \mathrm{F}\) ? (The density of water is \(1.00 \mathrm{~g} / \mathrm{mL}\).)

Short answer

To heat \(56\;\mathrm{L}\) of water from \(85{ }^\circ \mathrm{F}\) to \(212{ }^\circ \mathrm{F}\), the required heat is approximately \(39.24\;\mathrm{kJ}\).

Step-by-step solution

Convert temperatures to Celsius

Convert the initial and final temperatures from Fahrenheit to Celsius using the formula: \( T(\mathrm{C}) = \frac{5}{9}( T(\mathrm{F}) - 32 ) \). Apply this formula to both initial and final temperatures.

Calculate the mass of water

Calculate the mass of the water in grams using the density and volume. Since density is given as \(1.00\;\mathrm{g/mL}\), and there are \(1000\;\mathrm{mL}\) in a liter, the mass of water in grams is the volume in liters times \(1000\;\mathrm{mL/L}\).

Compute the temperature change

Calculate the change in temperature (\(\Delta T\)) in Celsius by subtracting the initial temperature in Celsius from the final temperature in Celsius.

Calculate the heat required (Q)

Use the specific heat formula, \( Q = mc\Delta T \), with \( m \) representing the mass of water in grams, \( c \) being the specific heat capacity of water (\( 4.184\;\mathrm{J/g\cdot\degree C} \)), and \( \Delta T \) the change in temperature in Celsius. Calculate \( Q \) to find the heat in joules.

Convert the heat from joules to kilojoules

Since \(1\;\mathrm{kilojoule} = 1000\;\mathrm{joules}\), divide the energy in joules by \(1000\) to convert it to kilojoules.

Key concepts

Specific Heat Capacity

Specific heat capacity, denoted by the symbol 'c', is a property that measures how much energy is required to raise the temperature of a substance by one degree Celsius. Defined as the amount of heat energy needed to change the temperature of one gram of a substance by one degree Celsius, it's a key concept in heat calculation chemistry. The specific heat capacity is unique to each material and is critical in determining how much heat energy, or 'Q', is required to change a substance's temperature.

For instance, water has a high specific heat capacity of approximately \(4.184 \mathrm{J/g\cdot\degree C}\), which means it requires a lot of energy to heat up. This explains why boiling a pot of water takes noticeable time and energy. In the context of the exercise, understanding that water has a high specific heat capacity allows us to predict that heating 56 liters from 85°F to 212°F will require a substantial amount of energy. The specific heat capacity is a pivotal factor in our final calculation of the heat energy needed.

Temperature Conversion

Temperature conversion between different scales is a fundamental skill in chemistry and various scientific applications. The most common scales are Celsius, Fahrenheit, and Kelvin. In this exercise, the temperatures initially provided in Fahrenheit must be converted to Celsius since most scientific calculations, including those relating to specific heat, are conducted in metric units.

The formula used for the conversion from Fahrenheit to Celsius is \( T(\mathrm{C}) = \frac{5}{9}( T(\mathrm{F}) - 32 ) \). This conversion is crucial to obtain the correct temperature change (\(\Delta T\)) in Celsius, which is needed to calculate the heat required to raise the temperature of the water. A common mistake is to overlook proper conversion, leading to incorrect calculations. Ensure that you apply this step carefully to obtain accurate results in a heat calculation scenario.

Density-Volume-Mass Relationship

The density-volume-mass relationship is a key concept in chemistry that describes how the mass of a substance relates to its volume and density. Density, typically represented by the Greek letter 'ρ' (rho), is the mass per unit volume of a substance and is usually expressed in grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³).

In mathematical terms, density is defined as \(\rho = \frac{m}{V}\), where 'm' stands for mass and 'V' stands for volume. In the context of heating water, knowing that the density of water is roughly 1.00 g/mL simplifies the calculation to find the mass, especially when the volume is given in liters. Since there are 1000 mL in a liter, to find the mass in grams, we simply multiply the volume of water by 1000. This clear relationship allows us to calculate the mass from volume conveniently, which is necessary for determining the heat required to increase the temperature of the water, as mass ('m') directly influences the heat equation \( Q = mc\Delta T \).