supercute5137
Use the known values of e_0, h, m, and y to calculate the following to five significant figures: hc (in eV nm), e^2(4πe0) (in eV nm), mc^2 (en keV), a0 (en nm) y E0 (en eV).
To calculate the given values, we need the following known values:
e_0 = 8.85418782 × 10^(-12) C^2/(N m^2) (vacuum permittivity)
h = 6.62607015 × 10^(-34) J s (Planck's constant)
m = 9.10938356 × 10^(-31) kg (electron mass)
y = 1.0973731568508 × 10^7 m^(-1) (Rydberg constant)
Now let's calculate the values:
1. hc (in eV nm):
hc = (h * c) / e (where c is the speed of light)
c = 299,792,458 m/s
e = elementary charge = 1.602176634 × 10^(-19) C
hc = (6.62607015 × 10^(-34) J s * 299,792,458 m/s) / (1.602176634 × 10^(-19) C)
hc = 1.2398419756476966 × 10^(-6) eV nm
Using five significant figures: hc = 1.2398 eV nm
2. e^2(4πe0) (in eV nm):
e^2(4πe0) = (e^2 * 4π * e_0)
= (1.602176634 × 10^(-19) C)^2 * 4π * 8.85418782 × 10^(-16) C^2/(N m^2)
e^2(4πe0) = 1.4401526687278486 × 10^(-15) eV nm
Using five significant figures: e^2(4πe0) = 1.4402 eV nm
3. mc^2 (in keV):
mc^2 = (m * c^2) / 1000 (to convert J to keV)
mc^2 = (9.10938356 × 10^(-31) kg * (299,792,458 m/s)^2) / 1000
mc^2 = 8.1871057768435 × 10^(-14) keV
Using five significant figures: mc^2 = 8.1871 keV
4. a0 (in nm):
a0 = (4πe_0 * h^2) / (m * e^2)
a0 = (4π * 8.85418782 × 10^(-12) C^2/(N m^2) * (6.62607015 × 10^(-34) J s)^2) / (9.10938356 × 10^(-31) kg * (1.602176634 × 10^(-19) C)^2)
a0 = 0.052917721092257136 nm
Using five significant figures: a0 = 0.052918 nm
5. E0 (in eV):
E0 = - (hc * R∞) / y
R∞ = 1.0973731568508 × 10^7 m^(-1) (Rydberg constant)
E0 = - (1.2398419756476966 × 10^(-6) eV nm * 1.0973731568508 × 10^7 m^(-1)) / 1.0973731568508 × 10^7 m^(-1)
E0 = - 1.2398419756476966 eV
Using five significant figures: E0 = -1.2398 eV
To calculate the values, we need the following known values:
- e_0: elementary charge = 1.602176634 x 10^-19 C
- h: Planck's constant = 6.62607015 x 10^-34 J s
- m: mass of an electron = 9.10938356 x 10^-31 kg
- y: fine-structure constant = 0.00729735257
1. Calculate hc (in eV nm):
hc = (h * c) / e (where c is the speed of light)
Plugging in the values, we get:
hc = (6.62607015 x 10^-34 J s * 2.998 x 10^8 m/s) / (1.602176634 x 10^-19 C)
hc ≈ 1.23984193 eV nm (rounded to five significant figures)
2. Calculate e^2(4πe0) (in eV nm):
e^2(4πe0) = (e^2 * 4 * π * e_0)
Plugging in the values, we get:
e^2(4πe0) = (1.602176634 x 10^-19 C)^2 * 4 * π * (8.8541878128 x 10^-12 F/m)
e^2(4πe0) ≈ 1.43996465 x 10^-9 eV nm (rounded to five significant figures)
3. Calculate mc^2 (in keV):
mc^2 = (m * c^2) / (1.602 x 10^-16 eV)
Plugging in the mass of an electron and the speed of light values, we get:
mc^2 = (9.10938356 x 10^-31 kg * (2.998 x 10^8 m/s)^2) / (1.602 x 10^-16 eV)
mc^2 ≈ 0.51099895 keV (rounded to five significant figures)
4. Calculate a0 (in nm):
a0 = (h^2 * y) / (4 * π * e_0 * m * c)
Plugging in the values, we get:
a0 = (6.62607015 x 10^-34 J s)^2 * 0.00729735257 / (4 * π * (8.8541878128 x 10^-12 F/m) * 9.10938356 x 10^-31 kg * (2.998 x 10^8 m/s))
a0 ≈ 0.05291772 nm (rounded to five significant figures)
5. Calculate E0 (in eV):
E0 = (m * (c^2)) / e
Plugging in the mass of an electron and the speed of light values, we get:
E0 = (9.10938356 x 10^-31 kg * (2.998 x 10^8 m/s)^2) / (1.602176634 x 10^-19 C)
E0 ≈ 0.51099895 eV (rounded to five significant figures)