Question

Water is an excellent coolant as a result of its very high heat capacity. Calculate the amount of heat that is required to change the temperature of \(10.0 \mathrm{~kg}\) of water by \(10.0 \mathrm{~K}\). Now calculate the kinetic energy of a car with \(m=1.00 \cdot 10^{3} \mathrm{~kg}\) moving at a speed of \(27.0 \mathrm{~m} / \mathrm{s}(60.0 \mathrm{mph}) .\) Compare the two quantities.

Short answer

Answer: The heat required to change the temperature of 10 kg of water by 10 K is 1.147 times the kinetic energy of the car moving at 27 m/s.

Step-by-step solution

Calculate the heat required to change the temperature of water

To calculate the heat required, we will use the formula Q = mcΔT, where m = 10 kg, c (specific heat capacity of water) = 4.18 J/(g·K) = 4180 J/(kg·K), and ΔT = 10 K. Q = (10 kg)(4180 J/(kg·K))(10 K) = 418,000 J So, the heat required to change the temperature of 10 kg of water by 10 K is 418,000 J.

Calculate the kinetic energy of the car

To calculate the kinetic energy of the car, we will use the formula KE = 0.5 mv^2, where m = 1.00 x 10^3 kg and v = 27 m/s. KE = 0.5 (1.00 x 10^3 kg)(27 m/s)^2 = 0.5 (1000 kg)(729 m^2/s^2) = 364,500 J So, the kinetic energy of the car is 364,500 J.

Compare the two quantities

Now we will compare the heat required to change the temperature of the water (418,000 J) and the kinetic energy of the car (364,500 J). 418,000 J / 364,500 J = 1.147 As 1.147 is greater than 1, we can conclude that the heat required to change the temperature of 10 kg of water by 10 K is 1.147 times the kinetic energy of the car moving at 27 m/s.

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity, commonly denoted as \(c\), is a property that tells us how much heat energy \(Q\) is needed to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). In the context of the exercise provided, water has a high specific heat capacity, \(4.18 \mathrm{J/(g\cdot K)}\) or \(4180 \mathrm{J/(kg\cdot K)}\), meaning it can absorb a lot of heat before its temperature rises substantially.

When calculating the amount of heat required to change the temperature of water (or any material), the formula \(Q = mc\Delta T\) is used, where:\(m\) is the mass of the water, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. This relationship ensures that the higher the specific heat capacity of a substance, the more energy it will store for a given mass and temperature change. Understanding this concept is crucial for a wide range of applications, from designing cooling systems to studying climate dynamics.

It's also essential to make note of the units used in calculations. Here, a common student mistake is to overlook converting grams to kilograms, which could lead to a thousand-fold error. Therefore, ensuring consistency in units across all formula parameters is a key step in accurate heat capacity calculation.

Calculating Kinetic Energy

Kinetic energy represents the energy an object possesses due to its motion. The formula to calculate the kinetic energy \(KE\) of an object is given by \(KE = \frac{1}{2} mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object.

For the car in the exercise, its motion is characterized by its mass and the speed at which it's traveling. By knowing these two variables, one can determine the kinetic energy, which is the work needed to accelerate the car from rest to that speed, and also the work required to bring the car to a stop from that speed.

In our example, the car has a mass of \(1.00 \times 10^3 \mathrm{kg}\) and is moving at a velocity of \(27.0 \mathrm{m/s}\). Using the kinetic energy formula, the energy is found to be \(364,500 \mathrm{J}\). This simple formula helps in understanding how mass and velocity are proportional to the energy in motion, with the velocity having a squared relationship indicating its higher influence on the kinetic energy.

Comparing Energy Quantities

Comparing different forms of energy allows us to understand their equivalence and potential transformations. In our example, we compared the heat energy required to increase the temperature of water to the kinetic energy of a car in motion.

The comparison shows that the heat required to raise the temperature of water by \(10 \mathrm{K}\) is slightly greater than the kinetic energy of the car. To be precise, it's \(1.147\) times larger. This ratio tells us that energy can exist in different forms - thermal energy in the water, and mechanical energy in the moving car - and we can actually compare their magnitudes.

Why is this comparison useful?

Comparing energies in differing forms make it possible to envisage energy conversion scenarios, like in an internal combustion engine where chemical energy is converted to kinetic energy. This sort of comparison underlines a foundational principle of physics: conservation of energy, emphasizing how energy may change form but its total quantity in an isolated system remains constant. Grasping this concept aids in understanding and applying energy principles in practical contexts, from engineering to environmental science.