which of the following statements best compares geothermal energy and the energy from natural gas based on their rates of renewal?
natural gas has a slower rate because any amount extracted is far lower than its generation inside earth.
natural gas has a faster rate because, being a carbon resource, it quickly replenishes as part of the carbon cycle.
geothermal energy has a faster rate because a vast amount of heat energy is always present inside earth's core.
geothermal energy has a slower rate because the water present below earth's surface cools down periodically.

Geothermal resources are reservoirs of hot water that exist at varying temperatures and depths below the Earth's surface. Mile-or-more-deep wells can be drilled into underground reservoirs to tap steam and very hot water that can be brought to the surface for use in a variety of applications, including electricity generation, direct use, and heating and cooling. In the United States, most geothermal reservoirs are located in the western states.

Renewable—Through proper reservoir management, the rate of energy extraction can be balanced with a reservoir's natural heat recharge rate.

Baseload—Geothermal power plants produce electricity consistently, running 24 hours per day / 7 days per week, regardless of weather conditions.

Domestic—U.S. geothermal resources can be harnessed for power production without importing fuel.

Geothermal energy has a faster rate because a vast amount of heat energy is always present inside the earth's core.

Explanation:

Geothermal energy is the energy that is produced from the earth's geosphere. The groundwater stored below the earth crust is heated by magma released by the earth's mantle. It comes out from earth crust from the hot spring. The natural gas is produced after the prolonged decomposition process of the organic matter generated from the dead plants and animals. Natural gas is a fossil fuel develop after millions of years by both biological and geological processes.

Therefore, rate of generation of geothermal energy is faster than that of the natural gas.

part 1: the half-life of the element is 22 years.

part 2: it will take 132.02 year for the sample to decay from 308.0 g to 4.8125 g.

explanation:

part 1: there are two ways to determine the half life time.

first way: the half life time can be determined directly from its definition as it is the time required for the sample to decay to its half concentration.

from the data the initial concentration was 45.0 g and decayed to its half value (22.5 g) in 22.0 years.

second way: the radio active decay is considered first order reaction so we can use the laws of first order reactions.

k = (1/t) ln (a/a-x), where k is the rate constant of the reaction,

t is the time of the reaction

a is the initial concentration of the reactant

a-x is the remaining concentration at time (t)

a = 45.0 g the initial concentration at t = 0 year.

at t = 22 year, a-x = 22.5 g

then k = (1/22 year) ln (45.0 g/ 22.5 g) = 0.0315 year-1.

the half-life time (t1/2) = ln 2/ k = 0.693/ k =0.693/ 0.0315 = 22 years.

part 2:

using the same law k = (1/t) ln (a/a-x),

using the given data; a = 308.0 g and a-x = 4.8125 g and the calculated k from the part 1 (k = 0.0315 year-1)

t = (1/k) ln (a/a-x) = (1/0.0315) ln (308.0/ 4.8125) = 132.02 years