Write down the value of the Boltzmann’s constant kb. It has the dimension of energy divided by temperature (T). At which value of T (in degrees Kelvin) is kBT equal to: 1 Joule, 1 Calorie, 1 electron volt, the ionization energy of the helium atom, the energy of one photon with a wave length of 5000 angstrom, the kinetic energy of one helium molecule with a velocity of 500 m/s? The first law of thermodynamics, AU = -W+Q, is a statement of unification of calorime- try with mechanics; that heat Q is a form of energy. W and Q are typically of the same order of magnitude in processes involving gases (like weather and gasoline engines), but for liquids and solids it is typically true that W < q.=”” this=”” disparity=”” is=”” the=”” reason=”” why=”” the=”” identification=”” of=”” heat=”” with=”” energy=”” was=”” obscured=”” until=”” the=”” 18th=”” century.=”” friction=”” does=”” not=”” heat-up=”” an=”” object=”” significantly=”” unless=”” you=”” start=”” doing=”” things=”” like=”” boring=”” canons=”” (rumfeld).=”” it=”” also=”” means=”” that=”” a=”” gasoline=”” engine=”” and=”” our=”” body=”” can=”” perform=”” lots=”” of=”” work=”” while=”” burning=”” (oridizing)=”” only=”” small=”” amounts=”” of=”” oil=”” or=”” chocolate.=”” here=”” are=”” a=”” few=”” more=””>
a. What is the value of the specific heat of water at room temperature and one atmosphere? What is the value of the latent heat of vaporization at the boiling point of water at one atmosphere? b. Imagine a bucket of water at T = 20° C. How fast do you have to throw the water from the bucket onto a wall for it to reach the boiling point on impact assuming the process is adiabatic and the water sticks and does not splash back? c. How much faster do you have to throw it for all the water to boil away into water-vapor on impact?
The Correct Answer and Explanation is :
he Boltzmann constant (k₆) is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the temperature of the gas.ts value is exactly 1.380649 × 10⁻²³ joules per kelvin (J/K).citeturn0search2
To determine the temperature (T) at which k₆T equals specific energy values, we can rearrange the equation k₆T = E to solve for T:
= E / k₆
Applying this formula:
- 1 Joule:
= 1 J / (1.380649 × 10⁻²³ J/K) ≈ 7.24297 × 10²² K - 1 Calorie:
ince 1 calorie = 4.184 joules, = 4.184 J / (1.380649 × 10⁻²³ J/K) ≈ 3.031 × 10²² K - 1 Electron Volt (eV):
eV = 1.602176634 × 10⁻¹⁹ joules, = 1.602176634 × 10⁻¹⁹ J / (1.380649 × 10⁻²³ J/K) ≈ 1.16045 × 10⁴ K - Ionization Energy of the Helium Atom:
he ionization energy of helium is approximately 79 eV, hich is 79 × 1.602176634 × 10⁻¹⁹ J ≈ 1.26572 × 10⁻¹⁷ J. = 1.26572 × 10⁻¹⁷ J / (1.380649 × 10⁻²³ J/K) ≈ 9.16832 × 10⁵ K - Energy of One Photon with a Wavelength of 5000 Ångström:
he energy (E) of a photon is given by E = hc/λ, here h is Planck’s constant (6.62607015 × 10⁻³⁴ J·s), is the speed of light (2.99792458 × 10⁸ m/s), nd λ is the wavelength in meters. or λ = 5000 Å = 5000 × 10⁻¹⁰ m = 5 × 10⁻⁷ m, = (6.62607015 × 10⁻³⁴ J·s × 2.99792458 × 10⁸ m/s) / 5 × 10⁻⁷ m 3.97289 × 10⁻¹⁹ J. = 3.97289 × 10⁻¹⁹ J / (1.380649 × 10⁻²³ J/K) ≈ 2.877 × 10³ K - Kinetic Energy of One Helium Molecule with a Velocity of 500 m/s:
he kinetic energy (KE) is given by KE = ½mv², here m is the mass and v is the velocity. he mass of a helium atom is approximately 4.002602 u, here 1 atomic mass unit (u) = 1.66053906660 × 10⁻²⁷ kg, o m ≈ 4.002602 × 1.66053906660 × 10⁻²⁷ kg ≈ 6.64648 × 10⁻²⁷ kg. E = 0.5 × 6.64648 × 10⁻²⁷ kg × (500 m/s)² 0.5 × 6.64648 × 10⁻²⁷ kg × 2.5 × 10⁵ (m²/s²) 8.3081 × 10⁻²² J. = 8.3081 × 10⁻²² J / (1.380649 × 10⁻²³ J/K) ≈ 6.017 × 10¹ K
hese calculations illustrate the temperatures at which the thermal energy (k₆T) corresponds to various specific energy values.t’s important to note that these temperatures are theoretical and serve to provide insight into the relationship between temperature and energy at the molecular and atomic levels.