What is the change in energy, Delta E, in kilojoules per mole of hydrogen atoms for an electron transition from n=5 to n=2?
delta E = 2.180 x 10^-18 J*(1/n1^2 - 1/n2^2) where n1<n2.
To calculate the change in energy (ΔE) for an electron transition from n=5 to n=2 in hydrogen atoms, we can use the Rydberg formula and convert the answer to kilojoules per mole.
The Rydberg formula is given by:
1/λ = R_H * (1/n₁² - 1/n₂²)
Where:
λ is the wavelength of the emitted or absorbed light,
R_H is the Rydberg constant for hydrogen (approximately 1.097373 × 10^7 m⁻¹),
n₁ is the initial principal quantum number (n=5),
and n₂ is the final principal quantum number (n=2).
First, we need to calculate the wavelength (λ) using the Rydberg formula.
1/λ = (1.097373 × 10^7 m⁻¹) * ((1/5²) - (1/2²))
Simplifying the expression:
1/λ = 1.097373 × 10^7 * (1/25 - 1/4)
1/λ = 1.097373 × 10^7 * (4/100 - 25/100)
1/λ = 1.097373 × 10^7 * (-21/100)
1/λ = -2.3045083 × 10^6 m⁻¹
Now, we can calculate the change in energy (ΔE) using the equation:
ΔE = hc/λ
Where:
h is the Planck constant (approximately 6.62607015 × 10⁻³⁴ J·s),
c is the speed of light (approximately 2.998 × 10^8 m/s),
and λ is the calculated wavelength.
Substituting the values into the equation:
ΔE = (6.62607015 × 10⁻³⁴ J·s * 2.998 × 10^8 m/s) / (-2.3045083 × 10^6 m⁻¹)
ΔE ≈ -8.534 × 10^(-19) J
To convert the energy from joules to kilojoules, divide by 1000:
ΔE ≈ -8.534 × 10^(-19) J / 1000
ΔE ≈ -8.534 × 10^(-22) kJ
Therefore, the change in energy (ΔE) for the electron transition is approximately -8.534 × 10^(-22) kilojoules per mole of hydrogen atoms.
To calculate the change in energy (ΔE) for an electron transition from one energy level to another, you can use the Rydberg formula:
1/λ = R((1/n1^2) - (1/n2^2))
Where:
- λ is the wavelength of light emitted or absorbed (in meters).
- R is the Rydberg constant, approximately 1.097 x 10^7 m^-1.
- n1 is the initial energy level.
- n2 is the final energy level.
In this case, the electron transition is from n = 5 to n = 2. Let's plug these values into the formula:
1/λ = R((1/5^2) - (1/2^2))
Simplifying further:
1/λ = R(1/25 - 1/4)
1/λ = R(1/25 - 1/4)
1/λ = R(4/100 - 25/100)
1/λ = R(-21/100)
Now we can solve for the wavelength, λ.
1/(-21/100) = λ
-100/21 = λ
λ ≈ -4.76 meters
Notice that the negative sign only indicates the direction of energy change (absorption or emission).
To convert this wavelength into energy change, we can use the equation:
E = hc/λ
Where:
- E is the energy change (in Joules).
- h is Planck's constant, approximately 6.626 x 10^-34 J·s.
- c is the speed of light, approximately 3.00 x 10^8 m/s.
Let's calculate the energy change:
E = (6.626 x 10^-34 J·s * 3.00 x 10^8 m/s) / -4.76 meters
E ≈ -4.19 x 10^-19 J
Finally, to convert this energy change into kilojoules per mole (kJ/mol), we need to divide it by Avogadro's number, which is 6.022 x 10^23.
ΔE = (-4.19 x 10^-19 J) / (6.022 x 10^23)
ΔE ≈ -0.070 kJ/mol
Therefore, the change in energy (ΔE) for an electron transition from n=5 to n=2 is approximately -0.070 kJ/mol of hydrogen atoms.