Question

How many Joules does it take to heat your body up by \(1^{\circ} \mathrm{C}\) if your (water-dominated) mass has a specific heat capacity of \(3,500 \mathrm{~J} / \mathrm{kg} /{ }^{\circ} \mathrm{C} ?\)

Short answer

Answer: To determine the energy required, we use the specific heat capacity formula: \(Q = mc\Delta T\). In this case, \(\Delta T = 1^{\circ} \mathrm{C}\), and \(c = 3,500 \mathrm{J}/\mathrm{kg}/^{\circ} \mathrm{C}\). However, the mass of the body is not given, so we can only express the energy required in terms of the mass (M): it will take \(3,500M\ \mathrm{J}\) to heat up the body by 1°C.

Step-by-step solution

Understanding Specific Heat Capacity Formula

The formula for specific heat capacity is given by \(Q = mc\Delta T\), where: - \(Q\) is the energy required in Joules, - \(m\) is the mass of the substance in kg, - \(c\) is the specific heat capacity in J/kg/°ç, and - \(\Delta T\) is the change in temperature in °C. We are given the specific heat capacity of the body, but we need the mass of the body along with the temperature change to determine the energy.

Define Temperature Change

In this case, we want to find out the energy required to heat the body by just 1°C, so \(\Delta T = 1^{\circ} \mathrm{C}\).

Mass of the Body

The mass of the body is not given in the problem. Since we cannot directly calculate the required energy without the mass, let's represent the mass by the variable \(M\). This way, we can express the energy in terms of \(M\).

Calculate Energy

Now that we know the specific heat capacity (\(c\)), the change in temperature (\(\Delta T\)), and have represented the mass with a variable (\(M\)), we can calculate the energy required using the formula: \(Q = mc\Delta T\) Plug in the given values and the variable: \(Q = M * 3,500 \mathrm{J}/\mathrm{kg}/^{\circ} \mathrm{C} * 1^{\circ} \mathrm{C}\) After multiplying, we get: \(Q = 3,500M \mathrm{~J}\) Since the mass of the body is not given in the problem, we can only express the energy required in terms of the mass (\(M\)): it will take \(3,500M\) Joules to heat up the body by 1°C.

Key concepts

Energy in Joules

When we discuss the energy involved in heating a substance, it is typically measured in joules. A joule is a unit of energy in the International System of Units (SI), and it is defined as the amount of work done when a force of one newton is applied over a distance of one meter. In the context of heating the human body, we are interested in calculating the amount of energy needed to increase the body's temperature.

Let's imagine you want to increase your body temperature by a single degree Celsius, and we know that your body's heat capacity is influenced by its water content. Considering the body is mostly water, if we have its mass, we can determine how many joules of energy are required to achieve the desired temperature rise. This explanation aligns with the solution provided where the energy, denoted by the variable 'Q', is represented as a function of mass. The formula becomes more tangible when you think of it in tangible terms: the fuel (energy in joules) necessary to warm up your 'biological engine' (your body) by a certain 'thermal increment' (temperature change).

Heat Transfer

Heat transfer is a fundamental concept in thermal physics that involves the movement of thermal energy from one place to another. It occurs due to temperature differences and follows the second law of thermodynamics, which states that heat flows naturally from an object at a higher temperature to an object at a lower temperature. There are three main modes of heat transfer: conduction, convection, and radiation. In the case of heating the human body, the heat transfer would primarily occur through conduction and convection as the warmer surroundings transfer energy to the body.

In our exercise, the calculation did not specify how the heat is transferred to the body, because the focus is on the amount of energy needed, irrespective of the transfer method. The specific heat capacity formula assumes that the necessary energy will be transferred to the body effectively, which is why we only need to find the energy in joules required for a temperature change, without detailing the heat transfer method.

Temperature Change

Temperature change is a crucial aspect of thermal processes and denotes the difference between the initial and final temperatures after heat transfer has occurred. It is often denoted with the symbol \(\Delta T\) in thermal physics equations. A positive temperature change indicates that an object has absorbed heat and become warmer, while a negative temperature change indicates that it has lost heat and become cooler.

In our specific exercise, we aimed to find the energy needed to induce a temperature increase of \(1^\circ \mathrm{C}\) in the human body. By stating a temperature change of \(1^\circ \mathrm{C}\), we've assumed a direct, linear relationship between energy added and the resultant temperature increase – a concept embedded in the specific heat capacity of the material, in this case, predominantly water. This means we expect a steady, uniform heating that doesn't vary across the different temperatures or phases of the substance.

Thermal Physics

Thermal physics is an area of study within physics that deals with the energy transfers and thermodynamic processes which induce changes in the temperature, phase, or internal energy of systems. It incorporates concepts like heat transfer, specific heat capacity, and temperature change to understand how substances interact with heat. Measuring the specific heat capacity of substances is crucial in this field as it helps predict how they will behave when energy is added or removed.

The exercise we've evaluated falls squarely within thermal physics, specifically dealing with the energy needed to change the temperature of a water-dominated mass, like the human body. By representing the energy required to heat the body with a formula derived from the principles of thermal physics, we acknowledge the subject's influence on real-world scenarios where understanding energy transfers can help in areas ranging from medical treatments to designing efficient heating systems.