Question

Find the energy difference between the ground state of hydrogen and deuterium (hydrogen with an extra neutron in the nucleus)

Short answer

Answer: To find the energy difference between the ground state of hydrogen and deuterium, follow these steps: 1. Calculate the reduced mass for hydrogen (µ_H) and deuterium (µ_D) using their respective nucleus masses (m_p for proton and m_d for deuteron) and the electron mass (m_e) in the formula: µ = (m_e * m_n) / (m_e + m_n) 2. Calculate the ground-state energy for hydrogen (E_H) and deuterium (E_D) using the formula: E = -13.6 * (µ / m_e) * (1^2 / 1^2) 3. Find the energy difference between the two ground states: Energy difference = E_D - E_H

Step-by-step solution

Find the reduced mass for each atom

First, let's calculate the reduced mass, µ, for hydrogen and deuterium. The reduced mass can be found using the following formula: µ = (m_e * m_n) / (m_e + m_n) where m_e is the mass of the electron (9.1093897 x 10^-31 kg), and m_n is the mass of the nucleus (proton or deuteron). For hydrogen, the mass of the proton (m_p) is 1.6726219 x 10^-27 kg. For deuterium, the mass of the deuteron (m_d) is approximately twice the mass of the proton (since it has one proton and one neutron), so m_d = 2 * m_p. Now, calculate the reduced mass for hydrogen (µ_H) and deuterium (µ_D) using the formula.

Calculate the ground-state energy for each atom

Using the ground-state energy formula, E = -13.6 * Z^2 / n^2, and the reduced mass ratio, ρ = µ_D / µ_H, we can calculate the energy difference between hydrogen and deuterium. First, calculate the ground-state energy of hydrogen (E_H) using the reduced mass for hydrogen (µ_H) and the formula: E_H = -13.6 * (µ_H / m_e) * (1^2 / 1^2) Next, calculate the ground-state energy of deuterium (E_D) using the reduced mass for deuterium (µ_D) and the formula: E_D = -13.6 * (µ_D / m_e) * (1^2 / 1^2)

Find the energy difference between the two ground states

Finally, find the difference between the two ground-state energies, E_D and E_H. Energy difference = E_D - E_H This will give us the energy difference between the ground state of hydrogen and deuterium.

Key concepts

Understanding the Reduced Mass Formula

The concept of reduced mass is essential when analyzing systems where two bodies, such as an electron and a nucleus, interact. It simplifies the problem by allowing us to treat the system as if it were a single body problem with an effective mass. The reduced mass formula is given by $$\mu =\frac{m_e\cdot m_n}{m_e+m_n}$$where $m_e$ is the mass of the electron, and $m_n$ is the mass of the nucleus.

For an electron circling a much more massive nucleus, the $m_n$ will be substantially larger than $m_e$, meaning the reduced mass will be very close to the mass of the electron. However, the slight difference can have a significant effect on energy calculations, which is why it is crucial to use this formula for more precise results. In the case of hydrogen and deuterium, the presence of an extra neutron in deuterium increases its nuclear mass and hence alters the reduced mass slightly compared to hydrogen.

Calculating Ground-State Energy

When quantum mechanics principles are applied, we can determine the ground-state energy of an atom. The ground-state energy, $E$, of an electron in a hydrogen-like atom can be calculated using the formula: $$E=-13.6\cdot \left(\frac{\mu }{m_e}\right)\cdot \left(\frac{Z^2}{n^2}\right)$$where $\mu$ is the reduced mass of the electron-nucleus system, $m_e$ is the mass of the electron, $Z$ is the atomic number, and $n$ is the principal quantum number.

For hydrogen ($Z=1$), the ground state ($n=1$) energy calculation incorporates the mass of the proton, while for deuterium, the calculation takes into account the combined mass of a proton and a neutron. It's this distinction in reduced mass that leads to a variation in ground-state energy between the two isotopes.

Comparison Between Hydrogen and Deuterium

Hydrogen and deuterium, while chemically similar, have key nuclear differences that affect their physical properties. Deuterium, or heavy hydrogen, contains an additional neutron. This additional mass means deuterium has a slightly different reduced mass when compared to regular hydrogen. This alteration changes the energy levels of the deuterium atom.

Using the ground-state energy formula, we find that the bound electron in deuterium experiences a slightly different energy level. The precise calculations provided in the step-by-step solutions reflect these differences: despite the ground-state energy being negative for both, due to the additional neutron, the absolute value is lower for deuterium. The resulting energy difference has implications for the spectral lines and chemical behavior, making deuterium identifiable and distinct from hydrogen.