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The root mean square velocity of the molecules in a sample of helium is 5/7th that of the moelcules in a sample of hydrogen. If the temperature of hydrogen sample is 0oC, then the temperature of the helium sample is about
- a)5oK
- b)0 K
- c)273oC
- d)100oC
Correct answer is option 'A'. Can you explain this answer?
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The root mean square velocity of the molecules in a sample of helium i...
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Given:
The root mean square velocity of the molecules in a sample of helium is 5/7th that of the molecules in a sample of hydrogen.
The temperature of the hydrogen sample is 0°C.
To find:
The temperature of the helium sample.
Solution:
To solve this problem, we need to use the root mean square velocity formula and the relationship between temperature and root mean square velocity.
Root Mean Square Velocity Formula:
The root mean square velocity (v) of a gas can be calculated using the formula:
v = √(3kT/m)
Where:
v = root mean square velocity
k = Boltzmann constant (1.38 × 10^-23 J/K)
T = temperature in Kelvin
m = molar mass of the gas
Relationship between Temperature and Root Mean Square Velocity:
For a given gas, the root mean square velocity is directly proportional to the square root of the temperature. Mathematically, this relationship can be expressed as:
v1/v2 = √(T1/T2)
Where:
v1 and v2 are the root mean square velocities of two gases.
T1 and T2 are the temperatures of the two gases.
Given:
The root mean square velocity of helium = (5/7) * root mean square velocity of hydrogen
Temperature of hydrogen = 0°C = 273K (conversion from Celsius to Kelvin)
Let's solve for the temperature of helium:
Using the given relationship between root mean square velocities:
v_helium/v_hydrogen = √(T_helium/T_hydrogen)
Substituting the given values:
(5/7) * v_hydrogen/v_hydrogen = √(T_helium/273)
Simplifying the equation:
(5/7) = √(T_helium/273)
Squaring both sides of the equation:
(5/7)^2 = T_helium/273
Solving for T_helium:
T_helium = (5/7)^2 * 273
Calculating the value:
T_helium ≈ 0°C
Therefore, the temperature of the helium sample is approximately 0°C, which corresponds to option 'A'.
The root mean square velocity of the molecules in a sample of helium is 5/7th that of the molecules in a sample of hydrogen.
The temperature of the hydrogen sample is 0°C.
To find:
The temperature of the helium sample.
Solution:
To solve this problem, we need to use the root mean square velocity formula and the relationship between temperature and root mean square velocity.
Root Mean Square Velocity Formula:
The root mean square velocity (v) of a gas can be calculated using the formula:
v = √(3kT/m)
Where:
v = root mean square velocity
k = Boltzmann constant (1.38 × 10^-23 J/K)
T = temperature in Kelvin
m = molar mass of the gas
Relationship between Temperature and Root Mean Square Velocity:
For a given gas, the root mean square velocity is directly proportional to the square root of the temperature. Mathematically, this relationship can be expressed as:
v1/v2 = √(T1/T2)
Where:
v1 and v2 are the root mean square velocities of two gases.
T1 and T2 are the temperatures of the two gases.
Given:
The root mean square velocity of helium = (5/7) * root mean square velocity of hydrogen
Temperature of hydrogen = 0°C = 273K (conversion from Celsius to Kelvin)
Let's solve for the temperature of helium:
Using the given relationship between root mean square velocities:
v_helium/v_hydrogen = √(T_helium/T_hydrogen)
Substituting the given values:
(5/7) * v_hydrogen/v_hydrogen = √(T_helium/273)
Simplifying the equation:
(5/7) = √(T_helium/273)
Squaring both sides of the equation:
(5/7)^2 = T_helium/273
Solving for T_helium:
T_helium = (5/7)^2 * 273
Calculating the value:
T_helium ≈ 0°C
Therefore, the temperature of the helium sample is approximately 0°C, which corresponds to option 'A'.
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The root mean square velocity of the molecules in a sample of helium is 5/7th that of the moelcules in a sample of hydrogen. If the temperature of hydrogen sample is 0oC, then the temperature of the helium sample is abouta)5oKb)0 Kc)273oCd)100oCCorrect answer is option 'A'. Can you explain this answer?
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