The molar heat capacity of a diatomic molecule is 29.1 J / K...

The molar heat capacity of a diatomic molecule is $29.1 \mathrm{J} / \mathrm{K} \cdot$ mol. Assuming the atmosphere contains only nitrogen gas and there is no heat loss, calculate the total heat intake (in $\mathrm{kJ}$ ) if the atmosphere warms up by $3^{\circ} \mathrm{C}$ during the next 50 years. Given that there are $1.8 \times 10^{20}$ moles of diatomic molecules present, how many kilograms of ice (at the North and South Poles) will this quantity of heat melt at $0^{\circ} \mathrm{C}$ ? (The molar heat of fusion of ice is 6.01 kJ/mol.)

Key Concepts

Latent Heat of Fusion

Latent heat of fusion is the energy required to change a substance from a solid phase to a liquid phase at its melting point without changing its temperature. This concept is key in understanding phase transitions and energy balances, especially when converting thermal energy into a measurable amount of melted material.

Unit Conversions in Thermodynamics

Unit conversions in thermodynamics ensure that all quantities are expressed in compatible units, such as converting energy from Joules to kilojoules or moles to mass in kilograms. Having a consistent unit system is critical for accurate calculations and meaningful comparisons between different energy quantities involved in a process.

Temperature Change

Temperature change (delta T) represents the difference in temperature between the initial and final states of a system. In thermodynamics, this value is crucial when calculating the heat absorbed or released by a substance, as it directly influences the amount of energy required for a given process.

Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin). This concept is fundamental in thermodynamics as it links the energy added to a system with the resulting temperature change and depends on the specific molecular structure and degrees of freedom of the substance.

Heat Energy Calculation

Heat energy calculation involves using the formula Q = n × C × ?T, where Q is the heat energy, n is the number of moles of the substance, C is the molar heat capacity, and ?T is the change in temperature. This relationship is essential when determining how much energy is absorbed or released during a temperature change in a system.

Transcript

00:01 For this problem, we will be determining the amount of ice that will melt at a certain heat.

00:09 Now, we first have to determine that heat, and that is based on two equations.

00:15 Well, determining the heat is based on one equation, and then finding the actual mass that will be melted is with another equation.

00:21 These two equations are q equals n, c, delta, t.

00:35 I'm sorry, that's supposed to be a delta and not an a.

00:42 And q equals n delta h.

00:49 So, if we say that the mueller heat of capacity for a diatomic element is 29 .1.

00:56 29 .1 is the value for the mueller heat of capacity.

00:59 The molar heat of capacity is c.

01:03 Q is just the heat, but the molar heat capacity is c.

01:10 So if we have the molar heat of capacity, and i'd say we also have the temperature difference that change, then and also the amount of moles present, we can determine the heat of the reaction.

01:25 So we will be using this first equation to determine that.

01:30 So q would equal the moles times the molar heat of capacity time to change in temperature.

01:38 Now, for the moles, we will have 1 .8 times 10 to the 20 moles.

01:51 And we'll be multiplying that by the molar heat of capacity, which we said was 29 .1 joules per kelvin per mole.

02:02 And then the change in temperature was a difference of 3 kelvin.

02:14 So if they gave us to us in celsius, we would know that if they said we increased 3 degrees, our final would be 3 degrees higher than our initial.

02:25 So it would be converting those celsius degrees into kelvin and then subtracting them.

02:32 But that step isn't necessary when you are doing a difference because converting between celsius and kelvin is just an addition.

02:40 So you will be subtracting what you are adding to the celsius anyway.

02:45 So if they tell you it's only increasing by 3 degrees celsius, that means it's also increasing by 3 kelvin...

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