Question

A large boulder whose mass is $1,000\mathrm{kg}$ having a specific heat capacity of $1,000\mathrm{J}/\mathrm{kg}/{}^{\circ }\mathrm{C}$ is heated from $0^{\circ }\mathrm{C}$ to a glowing $1,{800}^{\circ }\mathrm{C}$. How much more massive is it, assuming no atoms have been added or subtracted?

Short answer

Answer: The change in mass of the boulder is $2\times {10}^{-8}\mathrm{kg}$.

Step-by-step solution

Write the specific heat capacity formula

We can find the energy required to heat the boulder using the specific heat capacity formula, which states: $$Q=mc\mathrm{\Delta }T$$ where $Q$ is the heat energy, $m$ is the mass of the boulder, $c$ is the specific heat capacity, and $\mathrm{\Delta }T$ is the change in temperature (in this case, $1,{800}^{\circ }\mathrm{C}-0^{\circ }\mathrm{C}$).

Calculate the heat energy

Plugging the values into the specific heat capacity formula, we get: $$Q=(1,000\mathrm{kg})(1,000\mathrm{J}/\mathrm{kg}/{}^{\circ }\mathrm{C})(1,{800}^{\circ }\mathrm{C}-0^{\circ }\mathrm{C})$$ Solve for the heat energy, Q: $$Q=1,000\times 1,000\times 1,800\mathrm{J}$$ $$Q=1,800,000,000\mathrm{J}$$

Calculate the change in mass

Now, we know that energy and mass are related by Einstein's famous equation: $$E=m{c}^2$$ where $E$ is the energy, $m$ is the mass, and $c$ is the speed of light. We want to find the change in mass, which we can call $\mathrm{\Delta }m$. Rearranging the equation to solve for $\mathrm{\Delta }m$, we get: $$\mathrm{\Delta }m=\frac{E}{c^2}$$

Plug in the values and solve for the change in mass

Using the calculated heat energy (1,800,000,000 J) and the speed of light (approximately $3\times {10}^8\mathrm{m}/\mathrm{s}$), we can now find the change in mass: $$\mathrm{\Delta }m=\frac{1,800,000,000\mathrm{J}}{(3\times {10}^8\mathrm{m}/\mathrm{s}{)}^2}$$ $$\mathrm{\Delta }m=\frac{1,800,000,000}{9\times {10}^{16}}\mathrm{kg}$$ $$\mathrm{\Delta }m=2\times {10}^{-8}\mathrm{kg}$$

Conclusion

The boulder becomes more massive by $2\times {10}^{-8}\mathrm{kg}$ when it is heated from $0^{\circ }\mathrm{C}$ to $1,{800}^{\circ }\mathrm{C}$, assuming no atoms have been added or subtracted.

Key concepts

Understanding Thermodynamics in the Context of Heating a Boulder

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. The specific heat capacity of a material is a crucial thermodynamic property that quantifies the amount of heat required to change the temperature of a unit mass of a substance by a degree Celsius (or Kelvin).

In our exercise involving the heating of a boulder, thermodynamics governs the energy transfer as heat flows into the boulder, increasing its internal energy and hence its temperature. We've used the specific heat capacity to determine the amount of energy needed to raise the boulder's temperature from $0^{\circ }\text{C}$ to $1,{800}^{\circ }\text{C}$. The detailed calculation through the specific heat capacity formula incorporates these principles, ensuring the precise estimation of the heat energy, denoted as $Q$.

To optimize our understanding of this concept, we should visualize the process: Imagine the boulder absorbing heat energy, which causes the particles within it to vibrate more vigorously, leading to an increase in temperature. It's an exemplary scenario demonstrating how energy is transformed and transferred within a physical system, embodying the core ideas of thermodynamics.

Energy Conservation in the Heating Process

The law of energy conservation is a fundamental concept in physics stating that the total energy in an isolated system remains constant—it can neither be created nor destroyed, only converted from one form to another. This principle is at the heart of solving our boulder problem.

When we calculate the heat energy $Q$ absorbed by the boulder, we are applying the energy conservation principle. The energy doesn't disappear; rather, it transforms from the heat provided by the surrounding into the increased internal energy of the boulder, which we perceive as a rise in temperature. This conversion is described quantitatively by the specific heat capacity equation used in the step-by-step solution.

Enhancing this discussion with a practical view, consider energy conservation like a financial budget: just as you cannot spend more money than you have, the boulder cannot emit or absorb more energy than is transferred to it. Hence, the calculated energy $Q$ exactly equals the energy gain of the boulder, reflecting the concept of energy conservation in action.

Einstein's Mass-Energy Equivalence in Calculating Mass Change

Einstein's mass-energy equivalence is an iconic outcome of modern physics, encapsulated in the equation $E=m{c}^2$. It asserts that mass can be converted into energy and vice versa—unveiling a profound connection between these two seemingly disparate properties.

In the context of heating the boulder, Einstein's equation becomes relevant when quantifying the apparently negligible increase in mass due to the absorbed energy. Although the mass change $\mathrm{\Delta }m$ is minuscule ($2\times {10}^{-8}\text{kg}$), it illustrates a profound truth about our universe: energy has mass, and mass embodies energy. This deep insight expands beyond our exercise, touching the realms of nuclear reactions and cosmology, where mass-energy equivalence becomes most evident.

To put this in perspective: the mass change in our heated boulder is minute compared to everyday experiences because the speed of light $c$ in the equation is so large. Thus, even a stupendous amount of energy results in a slight change in mass. Nevertheless, this tiny mass increment is a testament to the transformation of absorbed thermal energy into the boulder's mass, a subtle yet real manifestation of Einstein's profound discovery.