Question

Consider the substances in Table 6.1. Which substance requires the largest amount of energy to raise the temperature of \(25.0 \mathrm{~g}\) of the substance from \(15.0^{\circ} \mathrm{C}\) to \(37.0^{\circ} \mathrm{C}\) ? Calculate the energy. Which substance in Table \(6.1\) has the largest temperature change when \(550 . \mathrm{g}\) of the substance absorbs \(10.7 \mathrm{~kJ}\) of energy? Calculate the temperature change.

Short answer

The substance requiring the largest amount of energy to raise its temperature from 15.0°C to 37.0°C is Substance X/Y/Z, with an energy requirement calculated as \(q_X\), \(q_Y\), or \(q_Z\). The substance with the largest temperature change when 550 g of the substance absorbs 10.7 kJ of energy is Substance X/Y/Z, with a temperature change of \(\Delta T_X\), \(\Delta T_Y\), or \(\Delta T_Z\).

Step-by-step solution

List the substances and their specific heat capacities

From Table 6.1, we have the following substances and their specific heat capacities (c): For unknown substances, replace X, Y, Z with their names: - Substance X: \(c_X = (c_X) \mathrm{~J/g^{\circ}C}\) - Substance Y: \(c_Y = (c_Y) \mathrm{~J/g^{\circ}C}\) - Substance Z: \(c_Z = (c_Z) \mathrm{~J/g^{\circ}C}\)

Calculate the energy required for each substance to raise its temperature from 15.0°C to 37.0°C

To find the energy required for each substance, given m=25 g and ΔT=(37-15)°C, we will calculate q using the formula: \(q = mc\Delta T\) For each substance: - Substace X: \(q_X = (25.0 \mathrm{~g})(c_X)(37.0 - 15.0)^{\circ}\mathrm{C}\) - Substace Y: \(q_Y = (25.0 \mathrm{~g})(c_Y)(37.0 - 15.0)^{\circ}\mathrm{C}\) - Substace Z: \(q_Z = (25.0 \mathrm{~g})(c_Z)(37.0 - 15.0)^{\circ}\mathrm{C}\)

Determine which substance requires the largest amount of energy

Compare the energy required for each substance (q_X, q_Y, and q_Z) and identify the substance requiring the largest amount of energy.

Calculate the temperature change when each substance absorbs 10.7 kJ of energy

To find the temperature change, we will rearrange the formula for q: \(\Delta T = \frac{q}{mc}\) Since q = 10.7 kJ and we need the value in J to be consistent with the specific heat capacities, we convert: \(q = 10.7 \mathrm{~kJ} \times 1000 \mathrm{~J/kJ} = 10700 \mathrm{~J}\). For each substance, given m=550 g and q=10700 J: - Substace X: \(\Delta T_X = \frac{10700 \mathrm{~J}}{(550 \mathrm{~g})(c_X)}\) - Substace Y: \(\Delta T_Y = \frac{10700 \mathrm{~J}}{(550 \mathrm{~g})(c_Y)}\) - Substace Z: \(\Delta T_Z = \frac{10700 \mathrm{~J}}{(550 \mathrm{~g})(c_Z)}\)

Determine which substance has the largest temperature change

Compare the temperature changes for each substance (ΔT_X, ΔT_Y, and ΔT_Z) and find the substance with the largest temperature change. Now you have found the substance that requires the largest amount of energy to raise its temperature from 15.0°C to 37.0°C, and the substance with the largest temperature change when 550 g of the substance absorbs 10.7 kJ of energy.

Key concepts

Energy Calculation

Calculating the energy required to change the temperature of a substance is crucial for understanding heat transfer in physical processes. In this context, the formula \[q = mc\Delta T\]is utilized, where:

• \(q\) is the heat energy in joules (J).
• \(m\) is the mass of the substance in grams (g).
• \(c\) is the specific heat capacity, indicating how much energy is needed to raise 1 g of the substance by 1°C.
• \(\Delta T\) is the temperature change in degrees Celsius (°C).

This formula helps us determine how much energy is needed to raise the temperature of a given mass of a substance by a certain temperature change. In practical scenarios, identifying the amount of energy involved is necessary for processes such as heating and cooling.
By substituting specific values for mass, specific heat capacity, and temperature change, we can calculate the specific energy required.

Temperature Change

Temperature change calculation is fundamental in thermal dynamics, especially when analyzing how substances react to energy input. The temperature change \(\Delta T\) is derived from the rearranged heat equation:\[\Delta T = \frac{q}{mc}\]Here:

• \(q\) is the energy added or removed, measured in joules (J).
• \(m\) is the mass of the substance in grams (g).
• \(c\) is the specific heat capacity.

By prioritizing energy and mass, you can determine how much the temperature of a substance will vary when a specific amount of energy is introduced. This concept is crucial in predicting and measuring temperature adjustments in everyday applications, such as cooking or industrial processes.Understanding this relationship ensures precise control in procedures requiring exact temperature settings.

Thermal Energy

Thermal energy refers to the total internal energy present in a substance due to the random motions of its particles. It plays a critical role in thermodynamics, as it governs heat transfer processes.The amount of thermal energy absorbed or released by a substance is directly connected to its specific heat capacity and the energy equation \(q = mc\Delta T\). Substances with higher specific heat capacities can absorb more energy without undergoing significant temperature changes, making them crucial in systems where temperature stability is required.Exploring how substances store and transfer thermal energy helps in various scientific and engineering practices, especially where heat retention or dissipation is needed.

Heat Transfer

Heat transfer is the movement of thermal energy from one body or region to another, driven by temperature differences. This process occurs through three primary modes, namely conduction, convection, and radiation. 1. **Conduction**: Heat moves through a material, with energy being passed from one particle to another, typically in solids. Metals, with their high thermal conductivity, facilitate conduction efficiently. 2. **Convection**: This mode involves the movement of fluid (liquid or gas) carrying heat along with it. It's observable in boiling water, where hot water rises and cold water sinks, creating a circular motion. 3. **Radiation**: Thermal energy is emitted as electromagnetic waves, such as infrared rays, without the need for a medium. This process allows the Sun's heat to reach Earth. Heat transfer is pivotal in various applications, from designing heat exchangers and building constructions to understanding natural phenomena. By mastering how thermal energy moves, we can optimize heating, cooling, and insulation strategies in multiple domains.