How Many Quantum States Exist in 500cm^3 of Water?

[v_pino]

Homework Statement

In lecture for stats and thermal, we briefly talked about quantum states and went through an example. However, the lecturer simply told us the answer to the second part of the question without going through it. Here's the question:

1a.) How many molecules of H2O are in 500cm^3 of water?

1b.) How many quantum states are there between zero and energy of this water?

Homework Equations

density = m/V

The Attempt at a Solution

Using density as 1000kgm^-3, I worked out the answer to be 1.67x10^26 molecules, which he said was correct.

He said that the answer to the second part is 4.56x10^28 , but I couldn't see why.

THanks.

 

[mark726]

Thank you for your post. In order to better understand the answer to the second part of the question, it is important to first understand what quantum states are and how they relate to energy levels. Quantum states refer to the different energy levels that a particle or system can exist in. In the case of water molecules, these energy levels are determined by the motion and arrangement of the atoms within the molecule.

Now, in order to determine the number of quantum states between zero energy and the energy of the water, we need to consider the energy levels that the water molecules can occupy. This can be calculated using the formula E = hν, where E is the energy, h is Planck's constant, and ν is the frequency of the energy. In this case, the energy of the water can be approximated as the energy of a single water molecule, which can be calculated using the formula E = 3kT/2, where k is Boltzmann's constant and T is the temperature.

Using the density you calculated (1000 kg/m^3) and the molar mass of water (18.015 g/mol), we can determine the mass of 500 cm^3 of water to be 9.0075 grams. Converting this to kilograms, we get 0.0090075 kg. Plugging this value into the formula for energy, we get E = 3(1.38x10^-23 J/K)(298.15 K)/2 = 6.137x10^-21 J.

Now, using the formula E = hν, we can solve for the frequency ν. Rearranging the equation, we get ν = E/h = (6.137x10^-21 J)/(6.626x10^-34 J*s) = 9.260x10^12 Hz. This represents the frequency of the energy of a single water molecule.

Finally, to determine the number of quantum states, we can use the formula N = n^3, where n is the number of energy levels. In this case, n can be approximated as the frequency ν divided by the energy of a single quantum state, which we calculated to be 9.260x10^12 Hz. Plugging these values into the formula, we get N = (9.260x10^12 Hz)^3 = 7.34x10^38 quantum states.

I hope this explanation helps you understand