Why does the narrator of "The Cask of Amontillado" by Edgar Allan Poe want revenge?

The narrator of "The Cask of Amontillado" wants revenge against his friend and fellow nobleman, Fortunato, because Fortunato has insulted him in some fashion. The exact details of this transgression are never named within the text; rather, the narrator only asserts, "The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge." With a soon-to-be murderer acting as our sole perspective in this story, it is hard to discern whether or not this insult was as grave as is implied. Is this a matter of actual slander? Or is the narrator simply paranoid, bloodthirsty, or mad? We are never given any definitive answer... That being said, Fortunato certainly does not seem to think he has committed any wrongdoings, as he greets the narrator "with excessive warmth." 

Nonetheless, the narrator assures us that "[a]t length [he] would be avenged," and proclaims proudly, "I must not only punish but punish with impunity" for "[a] wrong is unredressed when retribution overtakes its redresser... It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong." 

When does Lady Macbeth use flattery on Macbeth whilst persuading him to kill Duncan in Act 1, Scene 7?

In this particular scene, Lady Macbeth uses a great deal more insult that she does flattery to sway her husband.  However, after calling him a coward and saying that he's not really acting "like a man" at this point -- now that he's ambivalent about committing the murder of Duncan -- she does say, "When you durst do it, then you were a man; / And to be more than what you were, you would / Be so much more the man.  Nor time nor place / Did then adhere, and yet you would make both" (1.7.56-59).  In other words, she says that the decision Macbeth had made earlier, to kill the king, was a really manly decision, and he will be even more manly if he goes through with that decision now.  At the time, when he'd initially conceived of the plot to murder Duncan, nothing was ready -- it wasn't the time or the place -- and yet he still thought of and determined to enact this plan.  She seems to admire this and think it courageous and masculine.

Further, Lady Macbeth claims that if Macbeth can just pluck up the courage to move forward, there is no way that they will fail to succeed.  She says, "screw your courage to the sticking place / And we'll not fail" (1.7.70-71).  She makes it sound as though the whole plan hinges on him: if he can muster his courage then there is no possible way for things to go wrong.  This is relatively complimentary.

Why does water contract while increasing temperature from 0 to 4 degree C?

Water molecules are composed of one oxygen bound to two hydrogen atoms through a single covalent bond with each. These bonds are not evenly shared; oxygen has more positive charges in its nucleus and is able to exert a stronger pull on the electrons in the bond. Therefore, the electrons spend more of their time closer to the oxygen, which gives the oxygen a partial negative charge. The hydrogens, having less time around the electrons, do not have their single positive charge completely cancelled out, so they have a partial positive charge. We call this type of condition polarity. Water is one of the most polar molecules, and this explains a great deal about its properties.

The reason for the phenomenon observed when water is at 4 and 0 degrees is due both to the nature of this polarity, and to the behavior of molecules in their different phases. Liquids are typically higher in energy than their solid counterparts of the same substance under the same conditions. This means the liquid molecules are moving around more, and therefore will have a lower density. So, it shouldn't surprise us that as water cools, the trend is, on average, for the density to increase.

The polarity of water factors into the cooling as well. When water cools, the decreased motion of the molecules allows the molecules to "lock" themselves into the arrangement that best satisfies their polar charges; oxygens are near hydrogens, and vice versa. Due to the geometric shape of the water molecule, this results in a specific hexagonal crystal shape. These shapes are less dense, on average, than the liquid form of water because they don't have the energy to squeeze themselves into the less-favorable spaces.

When the water reaches 4 degrees, these shapes are beginning to take form, and the water is both increasing in density (as it cools) and decreasing in density (as the crystal shapes form and lock the molecules into a particular arrangement). By the time the water has reached 0 degrees Celsius, the locking effect is complete, and there's actually "wasted" space in the crystal that was once occupied by the slightly more energetic 4 degree water.

The molality of a 1L Solution with x percent H2SO4 is 9. The weight of solvent present in the solution is 910 grams. What is the value of x?

Molality is a measure of concentration defined as moles of solute per kg of solvent.  So if the molality of the sulfuric acid solution is 9, then that means that there are 9 moles of sulfuric acid per 1 kg of solvent.  So we have a solution that is 9 molal in concentration and the amount of solvent in it is 910 g which is the same as 0.910 kg.  So we multiply the two to get the moles of sulfuric acid in the solution.

(9 mole / kg) * 0.910 kg = 8.19 moles sulfuric acid

Multiply this by the molecular weight of sulfuric acid (98.079 g/mole) to convert into grams.

8.19 moles * (98.079 g / mole) =  803.3 g sulfuric acid

Now divide by the density of sulfuric acid (1.84 g/mL) to convert the grams into mL's.

803.3 g * (1 mL / 1.84 g) = 436.6 mL sulfuric acid

So we have 436.6 mL of sulfuric acid total.  Since we know that we have 1 L (the same as 1,000 mL) of total solution, we know that means that the percentage of the sulfuric acid in the solution can be calculated as shown below.

436.6 mL H2SO4 / 1,000 mL total solution = 0.437 = 43.7%

So we know that the value of x in the original question (percentage of H2SO4 in the solution) is 43.7%

I need to know who Hamlet really trusts. Who is loyal to Hamlet?

Hamlet really trusts Horatio and no one else. And Horatio is completely loyal to Hamlet. Hamlet confides everything to Horatio. Horatio is the only one to whom Hamlet has confided the substance of his meeting with the Ghost on the battlements and who knows that Claudius murdered Hamlet's father to seize the throne. In one scene Hamlet expresses his confidence in Horatio.

Horatio, thou art e'en as just a man
As e'er my conversation cop'd withal.
....................................................
Nay, do not think I flatter;
For what advancement may I hope from thee,
That no revenue hast but thy good spirits
To feed and clothe thee? Why should the poor be flatter'd?
No, let the candied tongue lick absurd pomp,
And crook the pregnant hinges of the knee
Where thrift may follow fawning. Dost thou hear?
Since my dear soul was mistress of her choice,
And could of men distinguish her election,
Sh'hath seal'd thee for herself. For thou hast been
As one, in suff'ring all, that suffers nothing;
A man that Fortune's buffets and rewards
Hast ta'en with equal thanks; and blest are those
Whose blood and judgment are so well commeddled
That they are not a pipe for Fortune's finger
To sound what stop she please. Give me that man
That is not passion's slave, and I will wear him
In my heart's core, ay, in my heart of heart,
As I do thee.       (Act III, Scene 2)

Hamlet expresses these sentiments to Horatio just before asking him to help observe the King during the course of the play-within-a-play. Hamlet wants to have a witness to verify his own observations of Claudius when it comes to the place where the actor playing the villain pours poison in the ear of the actor playing the sleeping King. It turns out that Hamlet hardly needs a witness, since Claudius makes such a scene that he brings the play-within-a-play to a halt as he flees the room. Horatio is convinced that the Ghost who appeared to Hamlet on the battlements was indeed the ghost of Hamlet's father and was telling him the exact truth.

Horatio acts like a true friend to Hamlet throughout the remainder of the play, and he proves his complete loyalty when he endeavors to commit suicide and join Hamlet in death at the very end. But Hamlet wrests the poisoned goblet out of his friend's hand and asks him to "absent himself from felicity" for a while in order to tell his story to the members of the court. Horatio is necessary for this function, since he is the only person still alive who knows the whole truth. Without him, everyone in the Danish court would assume that Hamlet had committed a wilful murder in an attempt to usurp the throne from Claudius. Horatio could not have been more loyal to Hamlet.

How are the themes of kindness and trust communicated in the story?

The themes of a piece of literature are implied messages the author conveys to the reader. In “Thank You, M’am” by Langston Hughes, the themes of kindness and trust are evident throughout the story.

The theme of trust is developed from the beginning of the story when Roger attempts to steal Mrs. Luella Bates Washington Jones’ “kitchen sink” pocketbook. Although, she has him in her grasp, she trusts him to pick up the purse and not run away. After she drags him back to her rooming house, the pair carry on a conversation during which Mrs. Jones reveals things from her past. She trusts Roger to understand her reasons for divulging her own desperate times.

The door to her room remains open while she instructs Roger to wash up, and when she goes behind the curtain to cook their meal. Her purse sits in plain sight so that if Roger chose to, he could grab it and run. He does not because at this point he wants to be trusted.

In another corner of the room behind a screen was a gas plate and an icebox. Mrs. Jones got up and went behind the screen. The woman did not watch the boy to see if he was going to run now, nor did she watch her purse which she left behind her on the day-bed. But the boy took care to sit on the far side of the room where he thought she could easily see him out of the corner of her eye, if she wanted to. He did not trust the woman not to trust him. And he did not want to be mistrusted now.

In addition, the theme of kindness runs throughout the story. Mrs. Jones decides not to call the authorities when Roger tries to steal her purse, instead she takes him home to teach him a lesson with her actions. She notices he is a hungry, neglected young man, which prompts her to encourage him to look presentable, and share her meager supper with him. This includes splitting a ten cent dessert. While she is preparing the meal, Roger offers to run to the store for anything she might need as he tries to reciprocate for her kindness towards him.

Her greatest kindness may have been the respect she shows him by not asking him questions about his home life so as not to embarrass him. Others would argue the utmost kindness occurred at the end of the story when Mrs. Luella Bates Washington Jones hands him money so he can buy the blue suede shoes he so desperately wants.

When they were finished eating she got up and said, “Now, here, take this ten dollars and buy yourself some blue suede shoes.  

He is rendered practically speechless by this act of kindness towards him.

The boy wanted to say something else other than “Thank you, m’am” to Mrs. Luella Bates Washington Jones, but he couldn’t do so as he turned at the barren stoop and looked back at the large woman in the door. He barely managed to say “Thank you” before she shut the door. And he never saw her again.

Explain why Amir's task is to not only atone for his own sins but for his father's as well in the novel The Kite Runner.

The protagonist in Khaled Hosseini's novel The Kite Runner, Amir, is motivated by guilt for most of his adult life for an act he committed in childhood. More specifically, it was an act he did not commit: he witnessed the rape of his best friend, Hassan, but did nothing to intervene. In fact, Amir violently rejected Hassan, desperately ashamed and discomfited by interacting with him. 

Later, Amir's guilt leads him to return to Afghanistan and rescue Hassan's son. However, it is not solely Amir's guilt that pushes him to this, some might say courageous, decision. As is revealed later in the text, Hassan is in fact Amir's half brother, fathered by Baba. In returning to Afghanistan for Sohrab, Amir recognizes Hassan's relationship with Baba as legitimate, complete with the responsibilities due to a family member in need.