Question

What are the differences between the $2s$ orbital and the $1s$ orbital of hydrogen? How are they similar?

Short answer

The 1s and 2s orbitals of a hydrogen atom are similar in having a spherically symmetric electron probability distribution. However, they differ in size, energy levels, and the number of radial nodes. The 1s orbital is smaller in size and has a lower energy level compared to the 2s orbital, which is larger in size and has a higher energy level. Additionally, the 1s orbital has no radial nodes, while the 2s orbital has one radial node, which separates the electron cloud into two distinct regions: an inner sphere and an outer sphere.

Step-by-step solution

Define the 1s and 2s orbitals

In a hydrogen atom, the 1s and 2s orbitals are electron cloud regions surrounding the nucleus, where electrons with certain energy levels and angular momenta are most likely to be found. These orbitals are described by the wave functions corresponding to the principal quantum number $n$ and the azimuthal quantum number $l$. For the 1s orbital, $n=1$ and $l=0$. While for the 2s orbital, $n=2$ and $l=0$.

Compare the sizes of the orbitals

The 1s orbital has a smaller size compared to the 2s orbital. The size of an orbital is determined by its principal quantum number (n), which also represents the energy level of the electron in that orbital. As the value of $n$ increases, the average distance of the electron from the nucleus increases. Since the 2s orbital has a higher principal quantum number ($n=2$), it is larger than the 1s orbital ($n=1$).

Compare the energy levels of the orbitals

The energy of an electron in a hydrogen atom depends on the principal quantum number (n). The higher the value of $n$, the higher the energy level of the electron in that orbital. In the 1s orbital, $n=1$, and in the 2s orbital, $n=2$. Thus, the electron in the 2s orbital has a higher energy level than the electron in the 1s orbital.

Compare the number of nodes

A node is a region in an orbital where the probability of finding an electron is zero. The number of nodes in an orbital depends on the principal quantum number ($n$) and the azimuthal quantum number ($l$). The mathematical formula for the number of radial nodes is $n-l-1$. For the 1s orbital, with $n=1$ and $l=0$, there are no radial nodes ($1-0-1=-1$ with a negative result meaning no radial nodes). For the 2s orbital, with $n=2$ and $l=0$, there is one radial node ($2-0-1=1$).

Compare their electron probability distributions

Both the 1s and 2s orbitals have spherically symmetric electron probability distributions, which means the probability of finding an electron is the same in all directions around the nucleus. However, their electron probability distributions differ in terms of the presence of radial nodes. While the 1s orbital has no radial nodes, the 2s orbital has one radial node, which separates the electron cloud into two distinct regions: an inner sphere and an outer sphere. In summary, the 1s and 2s orbitals of the hydrogen atom are similar in terms of having a spherically symmetric electron probability distribution. However, they differ in size, energy levels, and the number of radial nodes.

Key concepts

1s and 2s Orbitals

Exploring the differences and similarities between the 1s and 2s orbitals in a hydrogen atom helps to achieve a foundational understanding of quantum mechanics and atomic structure. To begin, both orbitals represent areas where there's a high likelihood of locating an electron around the nucleus, but they differ significantly in terms of size and energy.

The 1s orbital, being the closest to the nucleus with a principal quantum number of $n=1$, is smaller in size. This compact area means electrons in the 1s orbital are held more closely by the nucleus due to lesser energy compared to electrons in higher orbitals. In contrast, the 2s orbital has a principal quantum number of $n=2$, indicating not only a larger size but also a higher energy level. This allows for a greater distance between the nucleus and the electron within the 2s orbital.

Another key distinction lies in the existence of nodes. Nodes are regions within an electron cloud where the probability of finding an electron is zero. The 1s orbital has none, while the 2s orbital features a single radial node, signifying a shift in phase of the wave function of the electron. Effectively, this node splits the electron probability distribution of the 2s orbital into two regions - an inner and an outer one, characteristic of its higher energy state.

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom and are pivotal in distinguishing one orbital from another. Four quantum numbers - the principal ($n$), azimuthal ($l$), magnetic ($m_l$), and spin ($m_s$) quantum numbers - each provide unique information about an electron's position and movement.

For our 1s and 2s orbitals comparison, the principal ($n$) and azimuthal ($l$) quantum numbers are crucial. The principal quantum number indicates the electron's energy level and distance from the nucleus. It can be seen that as $n$ increases, so does the average distance from the nucleus, hence the increase in size and energy of the orbital.

The azimuthal quantum number $l$, which can range from 0 to $n-1$, determines the shape of the orbital and contributes to defining the angular momentum of an electron. In the case of s orbitals, $l$ is always 0; hence, both 1s and 2s orbitals are spherical in shape. The magnetic quantum number, which arises due to the orbiting electron's angular momentum in the presence of a magnetic field, and the spin quantum number, indicating the electron's intrinsic spin, are additional layers to an electron's quantum state but are not differentiated within the s orbitals as such.

Electron Probability Distribution

The electron probability distribution is a key concept when visualizing how electrons occupy certain regions around the nucleus. It is a quantifiable representation of where an electron is likely to be found. For the 1s and 2s orbitals, these distributions are spherically symmetric, suggesting that the electron is equally likely to be found at any angle around the nucleus.

However, the presence of a radial node in the 2s orbital introduces a layer of complexity to its probability distribution. While there is a high chance of locating an electron close to the nucleus in the 1s orbital, the 2s orbital has a region (node) where this probability drops to zero. Outside this node, the probability increases again, which is why the 2s orbital is often depicted with a denser outer sphere in illustrations.

The importance of understanding electron probability distribution extends beyond mere location. It influences the chemical properties of an atom, including its reactivity and the types of bonds it can form with others. Thus, comprehension of these distributions provides insight not just into the structure of an atom, but also into its interactions with its environment.