Tutor profile: Armaan N.
Subject: Organic Chemistry
$$Br_2$$ (liquid bromine) is a dark orange solution. Liquid bromine was added to both cyclohexane and cyclohexene, in 2 different tubes, but the scientist who performed this experiment forgot to label the tubes. The tube on the left turns from orange to colorless, while the tube on the right does not change color. You can assume that a color change indicates a reaction. Which tube, left or right, contains the solvent with the highest degree of saturation?
So this question contains two mini questions in disguise. The first is: which organic reagent, cyclohexane or cyclohexene, reacts with $$Br_2$$? The second is: which organic reagent, cyclohexane or cyclohexene, has the highest degree of saturation? To answer the first question: Cyclohexene reacts with bromine in the same way and under the same conditions as any other alkene. 1,2-dibromocyclohexane is formed. The reaction is an example of electrophilic addition. Cyclohexane does NOT react spontaneously with bromine. To answer the second question: Saturated hydrocarbons contain no double bonds, and so cyclohexane is the hydrocarbon with the highest degree of saturation. Now, to answer the final question, which tube has the solvent with the highest degree of saturation? It must be the one with cyclohexane, which we know does NOT react with liquid bromine spontaneously. Thus, it is the tube on the right.
Determine the domain of the function: $$f(x)=\ln(\arctan e^x)$$
To find the domain of a function, we want to find all of the allowed "x" values that will produce a real and measurable value of the function "f". Let's start with some basic restrictions we know. Off the bat, we know that the argument of a logarithmic function (i.e. anything inside the log) must be greater than zero. So, we write: $$\arctan e^x > 0$$ From here, we know that the output of the "arctan" function needs to be greater than zero. We know the range of the arctan function is $$(-\pi/2, \pi/2)$$, so we just need to ensure that the arctan function only spits out values greater than zero. Referencing the graph of arctangent, we see this happens when the argument of the arctan function, or in this case $$e^x$$, is greater than (but not equal to) zero, i.e..... $$e^x > 0$$ Trivially, we know this to be true for ALL values of x due to the nature of exponential functions. So, we conclude that the domain of the function is all real numbers. This might seem like a simple conclusion but be sure you understand WHY this is the case. As an extra challenge, now see what would happen if I challenged you to find the RANGE of the function. How might you go about finding that?
Perhaps the most surprising discovery involving the photoelectric effect was the observation of a threshold frequency that is independent of light intensity (or power). Please briefly describe the property or characteristic of light that is indicated by the existence of threshold frequencies. Give your answer in no more than two sentences.
Light energy can be described by a photon or a packet of energy, and that energy depends solely on the frequency (or wavelength) of the light. A single photon striking the surface of the metal is capable of ejecting one electron as long as the energy of the photon (which is proportional the frequency) equals or exceeds the work function of the metal.
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