Question

In terms of Rydberg constant \(R\). The wave number of first Balmer line is (A) \((5 \mathrm{R} / 36)\) (B) \((8 \mathrm{R} / 9)\) (C) \(\mathrm{R}\) (D) \((8 \mathrm{R} / 20)\)

Short answer

The wave number of the first Balmer line in terms of the Rydberg constant R can be calculated using the Rydberg formula for wave number \(\bar\nu = R \times ( \frac{1}{n_1^2} - \frac{1}{n_2^2})\). For the first Balmer line, \(n_1 = 2\) and \(n_2 = 3\). Plugging these values and simplifying the expression, we get \(\bar\nu = \frac{5R}{36}\), which corresponds to option (A).

Step-by-step solution

Recall the Rydberg formula for wave number

The Rydberg formula for wave number is given by: \(\bar\nu = R \times ( \frac{1}{n_1^2} - \frac{1}{n_2^2})\) Here, \(\bar\nu\) is the wave number, \(R\) is the Rydberg constant, \(n_1\) is the lower energy level, and \(n_2\) is the higher energy level.

Apply the formula for the first Balmer line

For the first Balmer line, we have \(n_1 = 2\) (as Balmer series transitions end at the second energy level) and \(n_2 = 3\) (as this is the initial energy level for the first Balmer line). Plugging these values into the Rydberg formula, we get: \(\bar\nu = R \times ( \frac{1}{2^2} - \frac{1}{3^2})\)

Simplify the expression

Simplify the expression obtained in step 2: \(\bar\nu = R \times ( \frac{1}{4} - \frac{1}{9})\) To find a common denominator for proceeding with the subtraction, we can use 36 as the denominator for both (as it is the least common multiple of 4 and 9) fractions: \(\bar\nu = R \times ( \frac{9}{36} - \frac{4}{36})\) Continue with the subtraction: \(\bar\nu = R \times \frac{9-4}{36}\) \(\bar\nu = R \times \frac{5}{36}\) Thus, the wave number of the first Balmer line in terms of the Rydberg constant R is: \(\boxed{\bar\nu = \frac{5R}{36}}\) This corresponds to option (A).

Key concepts

Balmer series

The Balmer series is a significant concept in the field of spectroscopy and atomic physics, specifically related to the hydrogen atom. It consists of a set of spectral lines that are produced when electrons in a hydrogen atom transition to the second energy level (n=2) from upper levels (n>2). This series is named after Johann Balmer, who was the first to derive an empirical formula for these visible light lines.

• Balmer series contains four prominent lines in the visible range: H-alpha, H-beta, H-gamma, and H-delta.
• These lines are crucial as they help in identifying elements in astronomical settings and laboratory spectra.
• The series can be observed when hydrogen gas is energized and emits light as the excited electrons fall back down to the n=2 level.

Understanding the Balmer series helps in comprehending how atomic emission spectra are produced. It is also foundational for more advanced study of atomic transitions and quantum mechanics.

Wave number

Wave number is an essential concept in electromagnetic theory and spectroscopy. It is defined as the number of wavelengths per unit distance, often used in describing waves of various kinds.Instead of using wavelength in spectroscopy, wave number is more frequently used because it directly relates to the energy of a wave. The wave number, represented as \(\bar{u}\), is calculated using the formula: \[ \bar{u} = R \times \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

• It is usually expressed in units of inverse meters (m⁻¹).
• Relates closely to the energy of photons, given by the equation \(E = h\bar{u}\), where \(h\) is Planck's constant.
• The concept of wave number helps to resolve fine details in spectral lines.

Understanding the wave number is important as it offers a different perspective and method of calculation regarding the properties of waves, especially when analyzing spectral lines.

Energy levels

Energy levels in atoms refer to the specific energies that electrons can have within an atom. Each energy level, or shell, is associated with a principal quantum number, denoted by \(n\). Electrons can move between these energy levels by absorbing or emitting photons of specific energies.

• The energy of an electron is quantized; this means it can only take discrete values defined by its energy level.
• Lower energy levels are closer to the nucleus, and higher levels are farther away.
• Transitions between these levels lead to absorption or emission of light, producing spectral lines.

Energy levels provide a fundamental explanation for the structure of the atom and the rules governing electron configurations. This concept is critical to both classical and quantum chemistry, as it indicates why atoms exhibit specific lines in their emission or absorption spectra.

Spectroscopy

Spectroscopy is a scientific technique used to measure and analyze the spectrum of light emitted, absorbed, or scattered by materials. It allows scientists to determine the composition, structure, and properties of matter by studying how it interacts with electromagnetic radiation.

• There are several types of spectroscopy, including atomic, molecular, and nuclear magnetic resonance (NMR) spectroscopy.
• Each type of spectroscopy provides unique information about the substance being examined.
• Spectroscopy plays a crucial role in astronomy, chemistry, and physics by providing a means to investigate properties of distant stars, planets, and chemical substances on Earth.

By using spectroscopy, scientists can also determine the electronic structure of atoms and molecules, which helps in understanding how matter behaves on a fundamental level. This area is indispensable for modern science and technology due to its wide range of applications.