Question

Three hundred thousand years after the Big Bang, the temperature of the universe was \(3000 \mathrm{~K}\). Because of expansion, the temperature of the universe is now \(2.75 \mathrm{~K}\). Modeling the universe as an ideal gas and assuming that the expansion is adiabatic, calculate how much the volume of the universe has changed. If the process is irreversible, estimate the change in the entropy of the universe based on the change in volume.

Short answer

Question: Estimate the change in the volume of the universe and its entropy, considering an adiabatic expansion and irreversible process, if the temperature changed from 3000 K to 2.75 K. Answer: The volume of the universe increased by a factor of approximately \(4.33 \times 10^5\), and the change in entropy is given by \(\Delta S = nR \ln{(4.33 \times 10^5)}\), where n is the number of moles of the gas and R is the universal gas constant.

Step-by-step solution

Write down the equation for adiabatic expansion

The adiabatic equation for an ideal gas is given by: \(T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}\) where \(T_1\) and \(T_2\) are the initial and final temperatures, \(V_1\) and \(V_2\) are the initial and final volumes, and \(\gamma\) is the heat capacity ratio, which is approximately \(5/3\) for a standard monoatomic ideal gas. We are given the initial and final temperatures, and we need to find the ratio of the final volume to the initial volume (\(V_2/V_1\)).

Solve for the ratio of the final volume to the initial volume

Using the adiabatic equation, we can solve for the ratio of the final volume to the initial volume (\(V_2/V_1\)): \(\frac{T_1}{T_2} = \frac{V_2^{\gamma - 1}}{V_1^{\gamma - 1}}\) Assuming the heat capacity ratio \(\gamma = 5/3\), this equation becomes: \(\frac{T_1}{T_2} = \frac{V_2^{2/3}}{V_1^{2/3}}\) Now, we can plug in the given temperatures (\(T_1 = 3000 \mathrm{~K}\) and \(T_2 = 2.75 \mathrm{~K}\)) and solve for the volume ratio: \(\frac{3000}{2.75} = \frac{V_2^{2/3}}{V_1^{2/3}}\)

Calculate the volume ratio

Now we can solve for \(V_2/V_1\): \(V_2^{2/3} = V_1^{2/3} \cdot \frac{3000}{2.75}\) Taking the cube root, we find the volume ratio: \(V_2 = V_1 \cdot \left(\frac{3000}{2.75}\right)^{3/2} \approx 4.33 \times 10^5 \cdot V_1\) So, the volume of the universe has increased by a factor of approximately \(4.33 \times 10^5\) since the initial temperature of \(3000 \mathrm{~K}\).

Estimate the change in entropy

Finally, we need to estimate the change in entropy for this irreversible process. For an ideal gas, the change in entropy can be given by: \(\Delta S = nR \ln{\frac{V_2}{V_1}}\) Here, \(n\) is the number of moles of the gas, and \(R\) is the universal gas constant. We can use the volume ratio calculated in step 3 to find the change in entropy: \(\Delta S = nR \ln{(4.33 \times 10^5)}\) The change in entropy depends on the number of moles, but the numerical value that multiplies \(nR\) gives us an estimation of the magnitude of the change in the entropy of the universe in this process.

Key concepts

Big Bang

The Big Bang theory is the prevailing cosmological model explaining the origin of the universe. According to the theory, the universe expanded from a hot, dense initial state nearly 13.8 billion years ago. This expansion can be viewed through the lens of physics as an adiabatic process, where the universe is thought to have expanded rapidly without exchange of heat with its surroundings.

As the universe expanded, it cooled down, leading to changes in the physical properties of the cosmic matter and energy. Initially, the universe was a plasma of particles and radiation; as it cooled, particles formed atoms, which eventually led to the formation of stars and galaxies. The temperature drop from 3000 K to a mere 2.75 K is indicative of this extensive expansion and cooling process over billions of years.

Entropy

Entropy is a fundamental concept in thermodynamics, often interpreted as a measure of disorder or randomness within a physical system. In an isolated system, like the universe, the second law of thermodynamics states that entropy can never decrease over time. This implies that processes occurring in the universe tend to increase its overall entropy.

Drawing on our example, the expansion of the universe results in an increase in the volume that its particles can occupy, spreading them out and increasing disorder. The irreversible nature of the universe's expansion contributes to a rising entropy value. Whether it's gases diffusing to fill a space or the cosmic background radiation, entropy is key to understanding the directionality and evolution of the universe.

Ideal Gas Law

The ideal gas law is a cornerstone of classical thermodynamics, expressed as the equation PV = nRT. Here, P is pressure, V is volume, n is the number of moles of the gas, R is the universal gas constant, and T is temperature. This equation is crucial in modeling the behavior of gases under various conditions, assuming the gases behave ideally, meaning they have no intermolecular forces and occupy no volume themselves.

When applied to cosmology, the ideal gas law can simplify the complex behaviors of the vast number of particles in the universe. Even though the universe is not a gas, this abstraction allows scientists to use the law as a baseline to estimate changes, like how its volume might change as it cools, just as we did in the exercise above.

Heat Capacity Ratio

The heat capacity ratio, often represented by the symbol \( \gamma \), is a dimensionless number that describes the ratio of the specific heat at constant pressure (\( C_p \) to that at constant volume (\( C_v \) for a given substance. For an ideal monoatomic gas, the heat capacity ratio is approximately 5/3. This value is central to understanding how a gas responds to an adiabatic process, such as expansion or compression, without the transfer of heat in or out of the system.

In our example, the heat capacity ratio is used in conjunction with the initial and final temperatures to determine how much the volume of the universe has increased during its adiabatic expansion. This ratio is fundamental in calculating how the temperature of the gas changes with its volume, which, as we've solved, results in a significant change in the scale of the universe from its infancy to the present era.