Question

If you used the 8400 kilojoules you expend in energy in one day to heat \(50000 \mathrm{~g}\) of water at \(20{ }^{\circ} \mathrm{C}\), what would be the rise in temperature? What would be the new temperature of the water?

Short answer

The temperature rise is 40.19°C. The new temperature is 60.19°C.

Step-by-step solution

Convert Energy into Appropriate Units

Convert the given energy expenditure from kilojoules (kJ) to joules (J).\[8400 \text{ kJ} = 8400 \times 1000 \text{ J} = 8400000 \text{ J}\]

Use the Specific Heat Formula

Apply the specific heat formula to find the temperature rise. The formula is: \[ Q = mc\triangle T\] where \( Q \) is the heat energy (8400000 J), \( m \) is the mass of the water (50000 g), and \( c \) is the specific heat capacity of water \( 4.18 \text{ J/g°C} \). Rearrange to find \( \triangle T \):\[ \triangle T = \frac{Q}{mc} = \frac{8400000}{50000 \times 4.18}\]

Calculate the Temperature Rise

Perform the calculation:\[ \triangle T = \frac{8400000}{50000 \times 4.18} = 40.19°C\]

Find the New Temperature

Add the temperature rise to the initial temperature of the water:\[ \text{New temperature} = 20°C + 40.19°C = 60.19°C\]

Key concepts

heat energy

Heat energy is the energy transferred from one body or system to another due to a temperature difference. When you heat an object, you are essentially transferring energy to it.
In the given exercise, you are using the 8400 kilojoules of energy you expend in a day to heat water. One important concept here is energy conversion. The original energy is given in kilojoules but we need it in joules for the calculations. Remember, 1 kilojoule equals 1000 joules. So, you multiply the kilojoules by 1000 to get joules, resulting in 8400 kJ * 1000 = 8400000 J.
Understanding how to convert energy into appropriate units is a crucial step in solving heat energy problems.

specific heat capacity

Specific heat capacity is a property of a material that indicates how much energy is required to raise the temperature of 1 gram of the material by 1°C. For water, the specific heat capacity is 4.18 J/g°C.
This property is essential for calculating how much energy is required to change the temperature of a substance. In the exercise, the formula to find the temperature change involves this property: \[ Q = mc\triangle T \] Here, Q is the heat energy in joules, m is the mass in grams, c is the specific heat capacity, and \(\triangle T\) is the temperature change.
Understanding this formula is crucial because it helps you relate the amount of energy transferred to the resultant temperature change in a given mass of material.

temperature rise

Temperature rise refers to the change in temperature that occurs when a substance absorbs heat energy. To find the temperature rise in the exercise, use the rearranged formula:
\[ \triangle T = \frac{Q}{mc} \] By substituting the given values: Q = 8400000 J, m = 50000 g, and c = 4.18 J/g°C, you get:
\[ \triangle T = \frac{8400000}{50000 \times 4.18} = 40.19°C \]
This means that the temperature of the water will increase by 40.19°C as a result of the energy added.
It's important to grasp this concept because it shows how energy input affects the temperature of an object and allows you to predict the resulting temperature changes.

joules

Joules are the standard unit of energy in the International System of Units (SI). One joule is defined as the energy transferred when one watt of power is applied for one second.
In the context of the exercise, the heat energy given is initially expressed in kilojoules, so you need to convert it to joules for calculations. This involves multiplying the kilojoules by 1000 (since 1 kJ = 1000 J).
Understanding the unit joules and being comfortable with unit conversions are fundamental skills in solving problems involving heat energy and temperature changes.