Question

If it takes \(526 \mathrm{J}\) of energy to warm \(7.40 \mathrm{g}\) of water by \(17^{\circ} \mathrm{C},\) how much heat would be needed to warm \(7.40 \mathrm{g}\) of water by \(55^{\circ} \mathrm{C} ?\)

Short answer

To find the heat needed to warm 7.40 g of water by 55°C, we can set up a proportion: \[ \frac{526 \, J}{17 \, °C} = \frac{Heat_2}{55 \, °C} \]. Solving for Heat2, we find that it takes approximately \(1705.88 \, J\) of energy to warm the water by 55°C.

Step-by-step solution

Understand the given information

We are given the following information: - It takes 526 J of energy to warm 7.40 g of water by 17°C. - We want to find how much heat is needed to warm 7.40 g of water by 55°C.

Set up a proportion

We can set up a proportion to solve this problem. The proportion will be in the form of: \[ \frac{Heat_1}{Temperature_1} = \frac{Heat_2}{Temperature_2} \] Where Heat1 and Temperature1 are the known values, and Heat2 and Temperature2 are the unknown values. In this case, we have: \[ \frac{526 \, J}{17 \, °C} = \frac{Heat_2}{55 \, °C} \]

Solve for Heat2

Now, we can solve for Heat2 by cross-multiplying and dividing: \[ Heat_2 = \frac{526 \, J \cdot 55 \, °C}{17 \, °C} \] Divide and calculate Heat2 to get the answer: \[ Heat_2 = 1705.88 \, J \]

Round the answer

As a final step, round the answer to two decimal places: \[ Heat_2 \approx 1705.88 \, J \] So, it will take approximately 1705.88 J of energy to warm 7.40 g of water by 55°C.

Key concepts

Understanding Heat Energy

Heat energy is a form of energy that can be transferred between substances, usually as a result of temperature differences. It is measured in units of joules (J) in the International System of Units. In the context of our textbook exercise, heat energy is the energy required to increase the temperature of water.

When you apply heat to a substance like water, it increases the motion of its molecules, resulting in a rise in temperature. The amount of heat needed to cause this temperature change depends on the mass of the water and its specific heat capacity, which is a measure of how much heat is required to raise the temperature of a certain mass of the substance by a specific amount. For water, the specific heat capacity is approximately 4.18 J/g°C.

So, in the textbook example, 526 J of energy raising the temperature of 7.40 g of water by 17°C indicates that the amount of heat energy required to achieve a particular temperature change can be calculated and further used to predict the heat needed for different temperature changes.

Proportional Reasoning in Chemistry

Proportional reasoning is a fundamental concept in chemistry, particularly when dealing with quantities and relationships between different variables. In the problem provided, we're using proportional reasoning to relate two different scenarios involving temperature changes and the corresponding heat energy required.

In simpler terms, if a certain amount of heat energy causes a specific temperature rise in a given mass of water, then a proportional amount of heat will cause a proportional temperature rise. This is based on the assumption that the specific heat capacity remains constant. The textbook solution shows how to set up and solve a proportion: \[ \frac{Heat_1}{Temperature_1} = \frac{Heat_2}{Temperature_2} \]Where the ratio of heat energy to temperature change for one scenario is equal to the ratio for the other scenario.

Using this method allows students to solve for an unknown variable by cross-multiplying and dividing the known values. This technique not only gives an exact number but also helps understand how different quantities in a system are interconnected.

Temperature Change Calculations

Temperature change in a substance refers to the difference in temperature as heat energy is added or removed. It is typically measured in degrees Celsius (°C) or sometimes in degrees Fahrenheit (°F). The more heat you add, generally, the greater the temperature change—assuming no phase change occurs, like melting or boiling, which requires additional energy.

In the example we've been given, the initial temperature change was 17°C, caused by adding 526 J of heat to 7.40 g of water. When asked to find out how much heat is needed to cause a 55°C temperature change in the same mass of water, we use the formula \[ Heat_2 = \frac{Heat_1 \times Temperature_2}{Temperature_1} \]to find that 1705.88 J of heat energy is required.

It's critical to understand that the temperature change is directly proportional to the amount of heat energy added, as long as the mass of the substance and its specific heat capacity remain constant. This allows us to calculate and predict the outcomes in various thermal processes, which is immensely useful in chemistry and other fields of science.